
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 the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with $a>0$ and $b>0$, the left and right foci are $F_{1}$ and $F_{2}$, respectively. A tangent line is drawn from point $F_{1}$ to the circle $x^{2}+y^{2}=a^{2}$, intersecting the right branch of the hyperbola at point $P$. It is also given that $\angle F_{1} P F_{2}=45^{\circ}$. Find the eccentricity of the hyperbola.

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