
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
A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with   $ x$  forints, where  $ 0<x<2C$ , uses the following strategy: if at a given time his capital is  $ y<C$ , he risks all of it; and if he has  $ y>C$ , he only bets  $ 2C\minus{}y$ . If he has exactly  $ 2C$  forints, he stops playing. Let  $ f(x)$  be the probability that he reaches  $ 2C$  (before going bankrupt). Determine the value of  $ f(x)$ .

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