
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 $F_1(-3,0)$, $F_2(3,0)$ are the two foci of the ellipse $\frac{x^2}{m} + \frac{y^2}{n} = 1$, and point $P$ is on the ellipse, find the value of $m+n$ when the area of $\triangle F_1PF_2$ is maximized for $\angle F_1PF_2 = \frac{2\pi}{3}$.

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