
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 \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola.

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