
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 \(\frac{x^{2}}{4}-\frac{y^{2}}{45}=1\) with two foci \(F_{1}\) and \(F_{2}\), let \(P\) be a point on the hyperbola. It is known that \(\left|P F_{2}\right|\), \(\left|P F_{1}\right|,\) and \(\left|F_{1} F_{2}\right|\) form an arithmetic sequence (or \(2\left|P F_{1}\right|=\left|P F_{2}\right|+\left|F_{1} F_{2}\right|\)), with the common difference being greater than 0. Find \(\angle F_{1} P F_{2}\).

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