
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 two foci $F_{1}$ and $F_{2}$, and $|F_{1}F_{2}|=2$, and a point $P$ on the ellipse such that the perimeter of $\triangle PF_{1}F_{2}$ is $6$.
(I) Find the equation of the ellipse $C$.
(II) If the coordinates of point $P$ are $(2,1)$, and a line $l$ passing through the origin $O$ intersects the ellipse $C$ at points $A$ and $B$, let $M$ be the midpoint of segment $AB$, $d$ be the distance from point $P$ to line $l$, and $M$, $O$, $P$ be collinear. Find the maximum value of $\frac{12}{13}|AB|^{2}+ \frac{13}{16}d^{2}$.

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