
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, with left and right foci $F\_1$ and $F\_2$. Draw a perpendicular line to the x-axis passing through $F\_2$, which intersects the ellipse at point $P$. If $\angle F\_1 P F\_2 = 45^{\circ}$, find the eccentricity of the ellipse, $e =$ ___.

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