
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 $F_1(-3,0)$ and $F_2(3,0)$ are the two foci of the ellipse $\frac{{{x}^{2}}}{m}+\frac{{{y}^{2}}{n}=1$, and $P$ is a point on the ellipse. When $∠F_1PF_2=\frac{2π}{3}$, the area of $\triangle F_1PF_2$ is maximum. Determine the values of $m$ and $n$.

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