
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
Let $F_1$ and $F_2$ be the two foci of the ellipse $x^2 + 4y^2 = 4$, and point $P$ is on the ellipse, with $F_1P \perpendicular PF_2$. Then, the area of $\triangle F_1PF_2$ is \_\_\_\_.

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