
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 the ellipse $x^{2}+ \frac {y^{2}}{b^{2}}=1(0 < b < 1)$, with its left and right foci being $F_{1}$ and $F_{2}$ respectively, and $|F_{1}F_{2}|=2c$. 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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