
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 $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ and the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ have coincident foci and reciprocal eccentricities. Let $F_1$ and $F_2$ be the left and right foci of the hyperbola $C$, and $P$ be any point on the right branch, calculate the maximum value of $\dfrac{|PF_2|}{|PF_1|^2}$.

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