
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
Fred and George play a game, as follows. Initially,  $x = 1$ . Each turn, they pick  $r \in \{3,5,8,9\}$  uniformly at random and multiply  $x$  by  $r$ . If  $x+1$  is a multiple of 13, Fred wins; if  $x+3$  is a multiple of 13, George wins; otherwise, they repeat. Determine the probability that Fred wins the game.

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