
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 eccentricity $\frac{3}{4}$, and the left and right foci are $F_{1}$ and $F_{2}$ respectively. The length of the minor axis is $2\sqrt{7}$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ Let $P$ be a point on the ellipse $C$ in the first quadrant. The lines $PF_{1}$, $PF_{2}$ intersect the line $x=5$ at points $A$ and $B$ respectively. Let the areas of triangles $PAB$ and $PF_{1}F_{2}$ be $S_{1}$ and $S_{2}$ respectively. If $\frac{S_{1}}{S_{2}}=\frac{9}{13}$, find $|AB|$.

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