Cus­tom The­or­ies

Guide writ­ten by Playspout and Snaeky. Con­tri­bu­tions from the Amaz­ing Com­munity.

This guide is cur­rently un­der­go­ing change. Keep in mind, strategies may change.

Feel free to use the gloss­ary as needed.

Cus­tom The­ory Ba­sics #

Cus­tom the­or­ies are the­or­ies made by play­ers in the com­munity. As of Septem­ber 3rd, 2022, there are 4 of­fi­cial cus­tom the­or­ies that con­trib­ute up to e150 \(\tau \) per the­ory; Wei­er­strass Sine Product made by Xelaroc (WSP), Se­quen­tial Lim­its by El­lip­sis (SL), Euler’s For­mula by Pea­nut, Snaeky, and XLII (EF), and Con­ver­gents to Square Root 2 (CSR2/​CS2) by Sol­arion. The the­or­ies will be ab­bre­vi­ated as WSP, SL, EF, and CSR2 from now on.

In or­der to bal­ance cus­tom the­or­ies with the main the­or­ies in the en­dgame, cus­tom the­or­ies have a low con­ver­sion rate from \(\rho\) to \(\tau\). WSP, SL, and CSR2 have con­ver­sion rates of \(\tau\) = \(\rho^{0.1}\) while EF has a \(\tau \) con­ver­sion rate of \(\tau\) = \(\rho^{0.4}\)

Which Cus­tom The­or­ies (CTs) should I do? #

In gen­eral, you want to be as ef­fi­cient as pos­sible since R9 does not af­fect cus­tom the­or­ies. If you can­not be act­ive, try not to do an act­ive the­ory or do an act­ive strategy. Some cus­tom the­or­ies are more act­ive than nor­mal the­or­ies and it is highly sug­ges­ted that if you are do­ing act­ive strategy for Cus­tom the­ory (SL be­fore all mile­stones, CSR2, or WSP) that you do an idle main the­ory (such as t2, t4, or t6) so that you don’t miss out on \(\tau/​hour\).

If you have time for act­ive strategies, try to do the CT with the highest act­ive \(\tau/​hour\). You can check this with the sim.

For idle time, do the one with the highest idle \(\tau/​hour \), (or the longest pub­lic­a­tion time if you’re do­ing overnights), with pref­er­ence to­ward EF and SL. For ex­ample, if SL has 2 \(\tau/​hour \) and CSR2 also has 2 \(\tau/​hour \), ideally we would pick SL. The reason we prefer SL and EF is be­cause these the­or­ies con­tain mul­tiple grow­ing vari­ables. This means the the­or­ies gen­er­ally re­quire less babysit­ting as the vari­ables grow by them­selves. The as­sump­tion of day­time idle is that we can check and pub­lish a the­ory every 2 hours or so. If you can only check every 8 hours idle, please see the overnight strategy just above.

Wei­er­strass Sine Product (WSP) #

WSP Over­view #

The very first of­fi­cial cus­tom the­ory; WSP was de­veloped by Xelaroc, who also came up with some of the strategies used in the the­ory. The idea be­hind the the­ory is to use the fac­tor­iz­a­tion of sine to in­crease \(\rho\). There are mul­tiple equa­tions with this the­ory, and some may look daunt­ing, so we’ll have a look at each one.

WSP Equa­tion De­scrip­tion #

\(\dot{\rho} = q_1^{1.04}q_2q\)

\(\dot{q} = c_2s_n({\chi}) / sin({\chi})\)

\(s_n({x}) := x\prod_{k=1}^{n}(1-\frac{x}{k\pi}^2)\)

\(\chi = \pi\frac{c_1n}{c_1+n/​3^{3}}+1\)

The first line states that the rate of change in rho is \(q_1^{1.04}q_2q\). Ini­tially it’s simply \(q_1q_2q\) without any ex­po­nent. With mile­stones we add more ex­po­nents.

For the second line, the higher the \(\chi\) (spelled ‘chi’, pro­nounced as ‘kai’), the higher the \(s_n({\chi})\). We want to in­crease \(\chi\) by in­creas­ing \(n \) and \(c_1\). The signs of \(s_n({\chi})\) and \(sin({\chi})\) will al­ways match, so the frac­tion can’t be neg­at­ive. Ad­di­tion­ally, the \(c_2 \) vari­able is a mile­stone which is not ini­tially avail­able.

The third line is the most com­plic­ated. Gen­er­ally we can fac­tor­ize an equa­tion when its graph touches the x-axis. For a sine curve, it touches the x-axis start­ing from x = 0, and re­peats every x= \(\pi\). These mul­ti­plied factors form the basis of the Wei­er­strass Sine Product. A sim­pler in­ter­pret­a­tion is that we can see ‘x’ ap­pear­ing both out­side and in­side the products in the nu­mer­ator. Since \(\chi\) is ‘x’ here, the higher the \(\chi\), the higher the \(s_n({\chi})\) as stated earlier.

Fi­nally, the ac­tual \(\chi\) equa­tion: in­creas­ing \(c_1 \) and \(n \) in­creases \(\chi\). Note that from the frac­tion, we don’t want to in­crease only \(c_1 \) or only \(n\). Rather we should in­crease both. Us­ing stand­ard strategies this should be no prob­lem. The \(n/​3^{3}\) part in the de­nom­in­ator is a mile­stone term. This means that \(n \) is bet­ter than \(c_1 \) as more \(n/​3\) mile­stones are ac­cu­mu­lated.

WSP Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

WSP strategy #

Early game the vari­able strengths are ordered as fol­lows:

\(q_2 \) ≈ \(c_2 \) > \(n \) > \(c_1 \) > \(q_1 \)

Late game these be­come:

\(n\) > \(q_2 \) ≈ \(c_2 \) > \(q_1 \) >>> \(c_1 \)

Idle

Be­fore you get e400 \(\rho \) for idle, simply auto­buy all.

Once you have e400 \(\rho \), \(c_1 \) starts to be­come ex­tremely bad. Be­cause of this, the new idle strategy would be to auto­buy all for 20 seconds or so. Then turn \(c_1 \) OFF. Con­tinue to auto­buy the rest of the vari­ables.

Act­ive

For a simple act­ive strategy be­fore e400 \(\rho \), simply auto­buy \(q_2 \) and \(c_2 \) since they double the rates long term. \(n \) and \(c_1 \) give ap­prox­im­ately 60% boost (with \(n \) be­com­ing more power­ful with mile­stones and vice versa for \(c_1\)). We will buy \(n \) and \(c_1 \) when their costs are less than 50% of the min­imum of \(q_2 \) and \(c_2 \).
For \(q_1 \), we will buy it when its cost is less than 10% of the min­imum of \(q_2 \) and \(c_2 \). For ex­ample, if \(q_1 \) costs 1.2e100 and \(q_2 \) costs 1e101, we would not buy \(q_1 \) as it’s ‘too ex­pens­ive’ com­pared to \(q_2 \).

For act­ive strategy, \(n \) starts to be­come more power­ful than \(q_2 \). If their costs are sim­ilar, we will pri­or­it­ize \(n \) first. For ex­ample, if \(n \) costs 1.4e101 and \(q_2 \) costs 1.2e101, we will buy \(n \) first. Sim­il­arly to the idle strategy, we will buy \(c_1 \) only for the first 20 seconds or so. If you want more in­form­a­tion on the dif­fer­ent strategies per­tain­ing to WSP, please see List of the­ory strategies

WSP mile­stone route #

All mile­stones into the 3rd/​last mile­stone. Then into 2nd mile­stone, then into 1st mile­stone.
For mile­stone swap­ping, swap all mile­stones from 2nd and 3rd into 1st mile­stone. Usu­ally you only do this when you’re about to pub­lish.

Se­quen­tial Lim­its (SL) #

SL Over­view #

SL, the second of­fi­cial cus­tom the­ory, uses a vari­ation of Stirl­ing’s for­mula to ap­prox­im­ate Euler’s num­ber (e≈2.71828). As up­grades are bought, the ap­prox­im­a­tion be­comes more pre­cise, in­creas­ing \(\dot\rho\) and \(\rho\) be­cause \(e-\gamma\) ap­proaches 0. As with the first of­fi­cial cus­tom the­ory (WSP), there are sev­eral equa­tions in this the­ory. Let’s ex­plore each one:

SL Equa­tion De­scrip­tion #

\(\dot{\rho}_1 = \frac{\sqrt{\rho_2^{1.06}}}{e - \gamma}\)

\(\gamma = \frac{\rho_3}{\sqrt[\rho_3]{\rho_3!}}\)

\(\dot{\rho_2} = a_1a_2a_3^{-ln{\rho_3}}\)

\(\dot{\rho_3} = b_1^{1.04}b_2^{1.04}\)

\(a_3 = 1.96\)

The first line is the main part of the equa­tion. We want to max­im­ize \(\dot{\rho_1}\) to in­crease \(\tau\). The ‘1.06’ ex­po­nent is from mile­stones. The de­fault is no ex­po­nent. From the equa­tion, we can see that \(\dot{\rho_1} \) is pro­por­tional to ap­prox­im­ately \(\sqrt{\rho_2} \). This means that if we quad­ruple \(\rho_2 \), we would ap­prox­im­ately double \(\rho_1 \) long term. The de­nom­in­ator of the frac­tion has a gamma sym­bol (\(\gamma\)) which looks like the let­ter ‘y’. As our \(\rho \) in­creases, our \(\gamma \) be­comes closer to ‘e’, so the de­nom­in­ator will de­crease, which in­creases \(\rho_1 \). We will ex­plore \(\gamma \) in the next equa­tion.

The second equa­tion refers to Stirl­ing’s ap­prox­im­a­tion of Euler’s num­ber ‘\(e\)’. As \(\rho_3 \) in­creases, \(\gamma \) con­verges to Euler’s num­ber. Long term we can ap­prox­im­ate this con­ver­gence as lin­ear. The im­plic­a­tion is if we double \(\rho_3 \), \(\gamma\) will be twice as close to Euler’s num­ber, so \(e-\gamma\) in the first equa­tion will be halved.

The third equa­tion relates \(\rho_2 \) with \(\rho_3 \) and some up­grades. The most in­ter­est­ing part is the ex­po­nent part con­tain­ing \(ln({\rho_3})\). The neg­at­ive ex­po­nent ac­tu­ally im­plies that as \(\rho_3 \) in­creases, \(\dot{\rho_2} \) DE­CREASES. If \(\rho_3 \) is high, \(\rho_2 \) does­n’t grow as fast (it still grows). This has im­plic­a­tion on the first equa­tion as well, since \(\dot{\rho_1} \) de­pends on \(\rho_2 \), which de­pends on \(\rho_3 \).

The fourth equa­tion relates \(\dot{\rho_3} \) with some up­grades. This one is re­l­at­ively simple; in­crease \(b_1 \) and \(b_2 \) to in­crease \(\rho_3 \). The ‘1.04’ ex­po­nents are from mile­stones.

The fi­nal equa­tion simply states the value of \(a_3 \). The lower the bet­ter. De­fault without mile­stone is \(a_3 = 2 \).

SL Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

SL strategy #

All vari­ables in SL are about the same in power, ex­cept for \(a_1\) and \(b_1\) (which are slightly worse than \(a_2 \) and \(b_2 \). Se­lect­ively buy­ing vari­ables at cer­tain times (act­ive) yields very little res­ults. There­fore, we can get away with auto­buy all for idle. Be­fore auto­buy, simply buy the cheapest vari­able. If you want more de­tails on SL strategies, in par­tic­u­lar the ex­e­cu­tion of vari­ous strategies, please see List of the­ory strategies.

Mile­stone swap­ping - why it works #

For act­ive, there is a mile­stone swap­ping strategy that is sig­ni­fic­antly faster than id­ling (ap­prox­im­ately twice the speed). If we care­fully ex­am­ine the ef­fects of each mile­stone, we can con­clude the fol­low­ing:

1st mile­stone: In­creases \(\rho_2 \) ex­po­nent and in­creases \(\dot{\rho_1} \) straight away. The ac­tual value of \(\rho_2 \) does not in­crease.
3rd/​4th mile­stone: In­crease \(b_1 \)/\(​b_2 \) ex­po­nents, and \(\dot{\rho_3} \), and \(\rho_3 \). This also in­creases \(\dot{\rho_1} \). However, the ef­fect is long-term and not in­stant­an­eous un­like the ef­fect of the 1st mile­stone.

We have dif­fer­ent mile­stones which af­fect the same thing (\(\dot{\rho_1} \)), but one is in­stant­an­eous, while the other builds over time. This forms the basis of ‘mile­stone swap­ping’, swap­ping mile­stones at cer­tain times to max­im­ize \(\rho_1 \) per hour. If you’ve done T2 mile­stone swap­ping, this should be fa­mil­iar.

We ini­tially put our mile­stones in the 4th and 3rd mile­stones. Once our \(\rho_3 \) does­n’t in­crease quickly any­more, we switch mile­stones to the 1st one to gain a burst of \(\dot{\rho_1} \). Once our \(\rho_1 \)is not in­creas­ing quickly any­more, we switch back to the 4th and 3rd mile­stone!

Mile­stone Swap­ping Strategies #

(Cour­tesy of Gen).

x>x>x>x rep­res­ent the max buy or­der of mile­stones not the amount al­loc­ated. For ex­ample, 4>3>1>2 means “Al­loc­ate everything into 4th mile­stone, then use leftovers into 3rd mile­stone, then into 1st mile­stone, then into 2nd mile­stone”.

From e75-e100 is 4>3>1>2 (60s) <-> 1>2>4>3 (60s)

SLMS2 is 1>2>4>3 (30s) --> 2>1>4>3 (60s) --> 1>2>4>3 (30s) --> 4>3>1>2 (60s), with \(b_1 \)\(b_2 \) off dur­ing the first two, and \(a_1 \)\(a_2 \) off dur­ing the last two

SLMS3 is 2>1>4>3 (20s) <-> 4>3>1>2 (60s)

When to Use Strategies un­til e100: SLMS
e100 - e175: SLMS (100-175)
e175 - e200: SLMS3
e200 - e300: SLMS

(note that it de­pends also on the swap­ping dur­a­tions, on the last range SLMS should be run with 60s [4/​3/​1/​2] and 20s on [1/​2/​4/​3] to be best). So from e200-e300, SLMS 4>3>1>2 (60s) <-> 1>2>4>3 (20s)

Post e300+ \(\rho\) #

At this point, the the­ory be­comes very idle. We simply auto­buy all vari­ables. Pub­lish at ap­prox­im­ately 8-10 mul­ti­plier. If you wish to im­prove ef­fi­ciency, you can dis­able \(a_1 \)\(a_2 \) at about 4.5 pub­lic­a­tion mul­ti­plier and \(b_1 \)\(b_2 \) at 6.0 mul­ti­plier un­til pub­lish.

SL mile­stone route #

Idle

Act­ive

Mile­stone Swap­ping (act­ive)

How to read nota­tion: 4/​3/​1/​2 means put all points into 4th mile­stones, use leftovers into 3rd mile­stones, etc.

SLMS is 4/​3/​1/​2 (60s) <-> 1/​2/​4/​3 (60s)

SLMS2 is 1/​2/​4/​3 (30s) --> 2/​1/​4/​3 (60s) --> 1/​2/​4/​3 (30s) --> 4/​3/​1/​2 (60s), with \(b_1 \)\(b_2 \) off dur­ing the first two, and \(a_1 \)\(a_2 \) off dur­ing the last two

SLMS3 is 2/​1/​4/​3 (20s) <-> 4/​3/​1/​2 (60s)

When to Use Strategies un­til e100: SLMS

e100 - e175: SLMS2

e175 - e200: SLMS3

e200 - e300: SLMS

Euler’s For­mula (EF) #

EF Over­view #

This cus­tom the­ory, along with Con­ver­gents to Square Root 2, were re­leased at the same time and is based on Euler’s For­mula of

\(e^{i*\theta} = cos{\theta} + isin{\theta}\), where ‘i’ is the com­plex num­ber.

EF is unique in that all the mile­stone paths are locked, so there’s no choice in which mile­stones to take. This was de­lib­er­ately done to pre­vent mile­stone swap­ping strategies and to bal­ance the the­ory. Fur­ther­more, the \(\rho\) to \(\tau \) con­ver­sion for this the­ory is uniquely at \(\rho^{0.4} \) rather than the usual \(\rho^{0.1} \) mean­ing that less \(\rho \) is needed to get an equi­val­ent amount of \(\tau\). Due to the con­ver­sion rate, EF can feel ex­tremely slow in com­par­ison to other the­or­ies, but it is the fast­est the­ory to e150 \(\tau \) and has the largest in­stant­an­eous jump in \(\tau \) out of all cus­tom the­or­ies.

EF Equa­tion De­scrip­tion #

\(\dot{\rho} = (a_1a_2a_3)^{1.5}\sqrt{tq^2+R^2+I^2}\)

\(G(t) = g_r+g_i\)

\(g_r = b_1b_2­cos{(t)}, g_i = ic_1c_2sin{(t)}\)

\(\dot{q} = q_1q_2\)

\(\dot{R} = (g_r)^2, \dot{I} = -(g_i)^2\)

The first line is the main equa­tion. We want to max­im­ize \(\dot{\rho_1} \). All the \(a_n \) terms and their ex­po­nents are ob­tained from mile­stones. Parts of the square root term are also ob­tained from mile­stones. Note that the \(R^2 \) and the \(I^2 \) terms are ef­fect­ively re­dund­ant at all stages of this the­ory; but due to them pur­chas­ing \(a_2 \) and \(a_3 \) re­spect­ively, they are very im­port­ant.

The second line defines the graph shown. Since \(G(t)\) is graphed on the com­plex over time, it is pos­sible to have it show as a particle spiral­ing through space.

The third line de­scribes \(g_r \) and \(g_i \), which are used to gen­er­ate ‘\(R\)’ and ‘\(I\)’ cur­ren­cies. This line by it­self does­n’t do much.

The fourth line simply de­scribes \(\dot{q} \). This is used in the first equa­tion dir­ectly.

The fifth and fi­nal line use the res­ults from the 3rd line, so ef­fect­ively \(\dot{R} = b_1^{2}b_2^{2}cos^2{(t)}\) and \(\dot{I} = c_1^{2}c_2^{2}sin^2{(t)}\)

EF Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

EF strategy #

Ini­tially, you only have \(\dot{t} \), \(q_1 \), and \(q_2 \) un­locked. Buy \(q_1 \) at about 1/​8th cost of \(q_2 \), and buy \(\dot{t} \) when it’s avail­able. At e20 \(\rho \) when auto­buy­ers are un­locked, for idle, simply auto­buy all. For act­ive, con­tinue to do what you were do­ing (buy­ing \(q_1 \) at 1/​8th cost of \(q_2 \)). There are also more ad­vanced strategies, in par­tic­u­lar EFAI. For its de­scrip­tion and ex­e­cu­tion, please see List of the­ory strategies.

The first 2 mile­stones are re­dund­ant by them­selves. The \(R^2 \) term and the \(I^2 \) term are in­sig­ni­fic­ant com­pared to the \(tq^2 \) term. Once you un­lock the 3rd mile­stone (\(a_1 \) term) however, we can buy \(a_1 \) at 1/​4th of \(q_2 \) cost.

EF mile­stone route #

Con­ver­gents to Square Root 2 (CSR2) #

CSR2 Over­view #

This cus­tom the­ory was re­leased at the same time as Euler’s For­mula. CSR2 is based on ap­prox­im­a­tions of \(\sqrt{2}\) us­ing re­cur­rent for­mu­lae. As the ap­prox­im­a­tions im­prove, the \(\dot{q}\) and \(\dot\rho\) im­prove, in­creas­ing \(\tau\). An ex­plan­a­tion of each sec­tion of the equa­tions is shown be­low:

CSR2 Equa­tion De­scrip­tion #

\(\dot{\rho} = q_1^{1.15}q_2q\)

\(\dot{q} = c_1c_2^2 |\sqrt{2} - \frac{N_m}{D_m}|^{-1}\), \(N_m = 2N_{m-1} + N_{m-2}, N_0 = 1, N_1 = 3\) \(D_m = 2D_{m-1} + D_{m-2}, D_0 = 1, D_1 = 2\) \(m = n + lo­g_2{(c_2)}\)

The first line is self ex­plan­at­ory. The ex­po­nents on \(q_1 \) are from mile­stones. ‘\(q\)’ will in­crease dur­ing the pub­lic­a­tion.

For the second line, both the vari­able \(c_2 \) and its ex­po­nents are from mile­stones. The ab­so­lute value sec­tion on the right de­scribes the ap­prox­im­a­tion of \(N_m \)/ \(D_m \) to \(\sqrt{2}\). As \(N_m \)/ \(D_m \) get closer to \(\sqrt{2}\), the en­tire right sec­tion gets lar­ger and lar­ger (be­cause of the -1 power).

The third and fourth lines are re­cur­rence re­la­tions on \(N_m \) and \(D_m\). This means that the cur­rent value of \(N_m \) and \(D_m \) de­pend on their pre­vi­ous val­ues. We start with \(N_0 \) = 1, \(N_1 \) = 3. The equa­tion will then read as:
\(N_2 \) = 2\(N_1 \) + \(N_0 \) -> \(N_2 \) = 2 x 3 + 1 = 7. Then \(N_3\) = 2\(N_2 \) + \(N_1 \) -> 2 x 7 + 3 = 17. Sim­ilar lo­gic is ap­plied to \(D_m \) equa­tions.

This oc­curs un­til we reach \(N_m \) and \(D_m \) reach whatever ‘m’ val­ues we have. This is shown in the next equa­tion:

The fourth equa­tion relates ‘m’ as de­scribed above. We can see that as we buy \(n \) and \(c_2 \), our \(m \) will in­crease, so the 2 re­cur­rence equa­tions above will ‘re­peat’ more of­ten and \(N_m \), \(D_m \) will in­crease. From how \(n \) and \(c_2 \) val­ues are cal­cu­lated, buy­ing 1 level of \(n \) or \(c_2 \) will in­crease \(m \) by 1.

CSR2 Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

CSR2 strategy #

Idle

For idle, we simply auto­buy all. The idle strategy does­n’t change much. If you’d like to be more ef­fi­cient while still be­ing idle, you can re­move mile­stones and stack them into the \(q\) ex­po­nent mile­stones when you’re about to pub­lish (from around e80 to e500). Don’t for­get to change mile­stones back after pub­lish­ing!

Once you have all mile­stones, auto­buy all!

Act­ive

The act­ive strategies are sig­ni­fic­antly more in­volved. De­pend­ing on how act­ive you’d like to be, there are sev­eral po­ten­tial strategies. There’s the stand­ard doub­ling chas­ing CSRd, which is just auto­buy all ex­cept \(c_1 \) and \(q_1 \), where you buy them when they are less than 10% cost of min­imum(\(c_2 \), \(q_2 \), \(n \)).

For the mile­stone swap­ping strategy, the gen­eral idea is to switch mile­stones from \(c_2 \) and its ex­po­nents, to \(q_1 \) ex­po­nent mile­stones whenever we are ‘close’ to a power­ful up­grade. Please see the The­ory Strategies sec­tion of the guide for how to per­form mile­stone swap­ping.

CSR2 Mile­stone Swap­ping Ex­plan­a­tion

This the­ory has a mile­stone swap­ping strategy be­fore full mile­stones. We have \(q_1 \) ex­po­nent mile­stones, which in­crease \(\dot\rho\) straight away. We also have \(c_2 \) re­lated mile­stones, which in­creases the \(q\) vari­able, which in­creases \(\dot\rho\).

The reason mile­stone swap­ping works is be­cause the be­ne­fits of us­ing \(c_2 \) re­lated mile­stones (hav­ing high \(q\)) re­main when you switch to \(q_1 \) ex­po­nent mile­stones. If we only use \(q_1 \) ex­po­nent, then we have really low \(q\). If we only use \(c_2 \) re­lated mile­stones, then we have high \(q\), but low \(\dot\rho\). If we reg­u­larly swap them, we can in­crease \(q\) through \(c_2 \) re­lated mile­stones, then take ad­vant­age of the \(q_1 \) ex­po­nent mile­stones, while keep­ing the high value of \(q\) we’ve ac­cu­mu­lated earlier!

For a more de­tailed ex­plan­a­tion on how to ac­tu­ally do the strategy, please see the The­ory Strategies sec­tion of the guide.

CSR2 mile­stone route #