
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)--(3,3)--(3,0)--cycle,dot); 
draw((3,3)--(0,0)--(0,1.5)--cycle,dot); 
draw((0,3)--(1,0),dot); 
MP("B",(0,0),SW);MP("A",(0,3),NW);MP("D",(3,3),NE);MP("C",(3,0),SE); 
MP("E",(0,1.5),W);MP("F",(1,0),S);MP("H",(2/3,2/3),E);MP("I",(4/7,12/7),N); 
dot((1,0));dot((0,1.5));dot((2/3,2/3));dot((4/7,12/7)); 
[/asy]
If $ABCD$ is a $3\times3$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is one-third from B to C along $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is ?

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