
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 $ABCD$ with side length 4, points $E$ and $H$ are on $\overline{AB}$ and $\overline{DA}$ respectively, such that $AE = 3$ and $AH = 1$. Points $F$ and $G$ are on $\overline{BC}$ and $\overline{CD}$, respectively, such that $BF = 1$ and $CG = 3$. Points $I$ and $J$ lie on $\overline{EH}$ such that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. If triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area 2, find $FI^2$.

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