
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 E: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a > b > 0$), its left and right foci are F<sub>1</sub> and F<sub>2</sub>, respectively. The distance from point F<sub>2</sub> to the line $x + \sqrt{3}y = 0$ is $\frac{1}{2}$. If point P is on the ellipse E, and the perimeter of triangle △F<sub>1</sub>PF<sub>2</sub> is 6,  
(1) Find the equation of ellipse E;  
(2) If a line l passes through F<sub>1</sub> and intersects ellipse E at two distinct points M and N, find the maximum value of the radius of the inscribed circle of triangle △F<sub>2</sub>MN.

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