
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
Square $ABCD$ has sides of length 2. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is a right isosceles triangle with $\angle AEF = 90^\circ$. A smaller square with one vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. Find the length of a side of this smaller square expressed as $\frac{a-\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime.

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