
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 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, such that $\triangle AEF$ is a right isosceles triangle (with $\angle AEF = 45^\circ$). A smaller square with vertex $B$ has sides that are parallel to those of $ABCD$, and one vertex on $\overline{AF}$. Find the length of a side of this smaller square and express it in the form $\frac{a - \sqrt{b}}{c}$, where $a, b, c$ are positive integers with $b$ not divisible by the square of any prime.

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