
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
Given an ellipse $$ x^2 + \frac{y^2}{b^2} = 1 $$ (0 < b < 1) with its left and right foci at F<sub>1</sub> and F<sub>2</sub>, respectively, and the distance between F<sub>1</sub> and F<sub>2</sub> is \( 2c \). If there exists a point P on the ellipse such that the distance from P to the line \( x = \frac{1}{c} \) is the arithmetic mean of \( |PF_1| \) and \( |PF_2| \), then the maximum value of b is ____.

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