
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively. The distance from point $F_{2}$ to the line $x+ \sqrt{3}y=0$ is $\frac{1}{2}$. If point $P$ is on ellipse $E$ and the perimeter of $\triangle F_{1}PF_{2}$ is $6$.

(1) Find the equation of ellipse $E$;
(2) If the line $l$ passing through $F_{1}$ intersects ellipse $E$ at two distinct points $M$ and $N$, find the maximum value of the radius of the inscribed circle of $\triangle F_{2}MN$.

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