
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
Petya and Vasya are playing the following game. Petya chooses a non-negative random value  $\xi$  with expectation  $\mathbb{E} [\xi ] = 1$ , after which Vasya chooses his own value  $\eta$  with expectation  $\mathbb{E} [\eta ] = 1$  without reference to the value of  $\xi$ . For which maximal value  $p$  can Petya choose a value  $\xi$  in such a way that for any choice of Vasya's  $\eta$ , the inequality  $\mathbb{P}[\eta \geq \xi ] \leq p$  holds? 

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