
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. Point $E$ is on $\overline{BC}$ such that $BE = 1$ and point $F$ is on $\overline{CD}$, so that $DF = 1$, forming $\triangle BEF$. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. There exists a point $G$ on $\overline{AE}$ such that $BG$ is perpendicular to $BE$. Find the length of a side of this smaller square and express your answer in simplest radical form.

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