
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>0,b>0)$ with eccentricity $\frac{\sqrt{2}}{2}$, and its left and right foci are $F_{1}$ and $F_{2}$ respectively. Point $P$ is a point in the coordinate plane such that $|\overrightarrow{OP}|=\frac{\sqrt{7}}{2}$ and $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=\frac{3}{4}$, where $O$ is the origin.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ A moving line $l$ passes through point $S(0,-\frac{1}{3})$ with slope $k$ and intersects the ellipse at points $A$ and $B$. Find the maximum intercept of the perpendicular bisector of chord $AB$ on the $x$-axis.

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