
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
Alice and Billy are playing a game on a number line. They both start at  $0$ . Each turn, Alice has a  $\frac{1}{2}$  chance of moving  $1$  unit in the positive direction, and a  $\frac{1}{2}$  chance of moving  $1$  unit in the negative direction, while Billy has a  $\frac{2}{3}$  chance of moving  $1$  unit in the positive direction, and a  $\frac{1}{3}$  chance of moving  $1$  unit in the negative direction. Alice and Billy alternate turns, with Alice going first. If a player reaches  $2$ , they win and the game ends, but if they reach  $-2$ , they lose and the other player wins, and the game ends. What is the probability that Billy wins?

*2018 CCA Math Bonanza Lightning Round #4.4*

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