
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
[asy] draw((0,0)--(0,3)--(4,3)--(4,0)--cycle,dot); draw((4,3)--(0,0)--(0,1.5)--cycle,dot); draw((0,3)--(2,0),dot); MP("B",(0,0),SW);MP("A",(0,3),NW);MP("D",(4,3),NE);MP("C",(4,0),SE); MP("E",(0,1.5),W);MP("F",(2,0),S);MP("H",(8/7,8/7),E);MP("I",(8/9,20/9),N); dot((2,0));dot((0,1.5));dot((8/7,8/7));dot((8/9,20/9)); [/asy]
If $ABCD$ is a $3\times4$ rectangle, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, compute the area of quadrilateral $BEIH$.

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