
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 $\frac{x^2}{2} - \frac{y^2}{b^2} = 1 (b > 0)$, its left and right foci are $F\_1$ and $F\_2$, respectively. One of its asymptotes is $y = x$, and the point $P(\sqrt{3}, y\_0)$ lies on the hyperbola. Determine the value of $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}$.

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