
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 $(C):\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and an eccentricity of $\frac{1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of the incircle of $\triangle F_{1}PF_{2}$ is $\frac{π}{3}$.

1. Find the equation of ellipse $(C)$;
2. Point $A$ is the left vertex of ellipse $(C)$, and a line with slope $k(k > 0)$ intersects $(C)$ at points $A$ and $M$. Point $N$ is on $(C)$, $MA⊥NA$, and $|AM|=|AN|$. Find the area of $\triangle AMN$.

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