
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
Let \( x, y \in \mathbf{R} \). Denote the minimum value among \( 2^{-x}, 2^{x-y}, 2^{y-1} \) as \( P \). When \( 0 < x < 1 \) and \( 0 < y < 1 \), what is the maximum value of \( P \)?

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