
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 an eccentricity of $\frac{1}{2}$, and its left and right foci are $F_{1}$ and $F_{2}$, respectively. A circle with center $F_{1}$ and radius $3$ intersects with a circle with center $F_{2}$ and radius $1$ at points on the ellipse $C$.
(I) Find the equation of the ellipse $C$;
(II) The line segment $PQ$ is a chord of the ellipse $C$ passing through point $F_{2}$, and $\overrightarrow{PF_{2}}=λ \overrightarrow{F_{2}Q}$.
(i) Find the perimeter of $\triangle PF_{1}Q$;
(ii) Find the maximum area of the incircle of $\triangle PF_{1}Q$ and the value of the real number $λ$ when this maximum area is obtained.

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