
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 common foci of an ellipse and a hyperbola, $P$ is their common point, and $\angle{F_1 P F_2}= \frac{2\pi}{3}$, the eccentricity of the ellipse is $e\_1$ and the eccentricity of the hyperbola is $e\_2$. Find the value of $\frac{3}{e\_1^{2}}+ \frac{1}{e\_2^{2}}$.

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