
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 the distance sum between point $P$ and the two foci $F\_1$, $F\_2$ of the hyperbola $x^{2}-y^{2}=1$ is constant, and the minimum value of $\cos ∠F\_1PF\_2$ is $-\frac{1}{3}$, determine the trajectory equation of point $P$.

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