
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
In the plane rectangular coordinate system $xOy$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$. Point $P$ lies on the ellipse $C$. Find the maximum value of the radius $r$ of the inscribed circle of $\triangle PF_{1}F_{2}$, and determine the coordinates of point $P$ if $\triangle POF_{2}$ is an isosceles triangle.

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