
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
We consider the following game for one person. The aim of the player is to reach a fixed capital  $C>2$ . The player begins with capital  $0<x_0<C$ . In each turn let  $x$  be the player’s current capital. Define  $s(x)$  as follows:  $$ s(x):=\begin{cases}x&\text{if }x<1C-x&\text{if }C-x<11&\text{otherwise.}\end{cases} $$  Then a fair coin is tossed and the player’s capital either increases or decreases by  $s(x)$ , each with probability  $\frac12$ . Find the probability that in a finite number of turns the player wins by reaching the capital  $C$ .

Present the answer in LaTex format: \boxed{Your answer}
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