
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 ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ (a > b > 0) and the hyperbola $\frac{x^{2}}{m^{2}}- \frac{y^{2}}{n^{2}}=1$ (m > 0, n > 0) share common foci $F_1$, $F_2$, and intersect at point P in the first quadrant. The eccentricities of the ellipse and hyperbola are $e_1$ and $e_2$, respectively. If $\angle F_{1}PF_{2}= \frac{\pi}{3}$, find the minimum value of $e_1 \cdot e_2$.

A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{\sqrt{3}}{2}$
D) $\frac{3}{2}$

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