
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
Given that $E$ is a point on the extension of the side $BC$ of the square $ABCD$, and a square $AEFG$ is constructed using $AE$ as its side such that points $A$ and $G$ are on the same side of $BE$. The extension of $BD$ intersects $AF$ at point $H$, with $HD = \sqrt{2}$ and $FH = 5\sqrt{2}$. What is the length of $BE$?

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