
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
Let square $ABCD$ have the side length of 2. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, creating an equilateral triangle $AEF$. A smaller square with vertex $B$ has its sides parallel to those of $ABCD$, and one vertex on $\overline{AE}$. Additionally, one side of this smaller square is tangent to a circle centered at $A$ with radius equal to $AE$. Find the side length, $s$, of this smaller square.

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