
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 a hyperbola and an ellipse with the same foci $F\_1$, $F\_2$, and a common point $P$. The eccentricity of the ellipse is $\sin \frac{\pi}{4}$. If $\cos \angle F\_1PF\_2 = \frac{1}{2}$, find the eccentricity of the hyperbola.

A) $2$
B) $\frac{\sqrt{5}}{2}$
C) $\frac{\sqrt{6}}{2}$
D) $\frac{\sqrt{7}}{2}$

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