
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 that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C$: $x^2 - y^2 = 2$, and point $P$ is on $C$ with $|PF\_1| = 2|PF\_2|$, find the value of $\cos∠F\_1PF\_2$.
A) $\frac{1}{4}$
B) $\frac{3}{5}$
C) $\frac{3}{4}$
D) $\frac{4}{5}$

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