
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
Sarah stands at  $(0,0)$  and Rachel stands at  $(6,8)$  in the Euclidena plane. Sarah can only move  $1$  unit in the positive  $x$  or  $y$  direction, and Rachel can only move  $1$  unit in the negative  $x$  or  $y$  direction. Each second, Sarah and Rachel see each other, independently pick a direction to move, and move to their new position. Sarah catches Rachel if Sarah and Rachel are every at the same point. Rachel wins if she is able to get  $(0,0)$  without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?

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