
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
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$. The line $l$ passing through the left focus $F_{1}$ intersects $C$ at a point $P$ in the first quadrant. If $|F_{1}P|=|F_{1}F_{2}|$ and the cosine value of the slope of line $l$ is $\frac{7}{8}$, then the eccentricity of $C$ is ______.

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