
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
In square $ABCD$, point $E$ lies on $\overline{AB}$ and point $H$ on $\overline{DA}$ such that $AE=AH.$ On $\overline{BC}$ lies point $F$ and on $\overline{CD}$ lies point $G$. Additionally, on diagonal $\overline{EH}$, points $I$ and $J$ are such that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. In this configuration, triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each have an area of $2$. Determine the value of $FI^2$ given the side length of square $ABCD$ is $4$.

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