
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
The road network graph of the city of Ikozapolis is equivalent to the edge network graph of an icosahedron. Jorgosz’s accommodation is located at one vertex of the icosahedron, while his favorite theater is located at the opposite vertex. After dark, when returning home from the theater, he becomes uncertain about which direction to take at each vertex. Assume that at each vertex, the probability that he meets someone who shows him the shortest route to his accommodation is $p$. Otherwise, he continues at random with no preferred direction, meaning it is possible he might backtrack. What should the value of $p$ be for there to be a 50% probability that he reaches his accommodation before returning to the theater?

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