
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 point P is an intersection point of the ellipse C_1 and the hyperbola C_2 with a common focus, where F_1 and F_2 are the left and right foci of the ellipse, and the eccentricities of the ellipse and the hyperbola are e_1 and e_2 respectively, if \overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0, calculate the value of \frac{1}{e_1^2} + \frac{1}{e_2^2}.

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