
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
Parallelepiped $ABCDEFGH$ is generated by vectors $\overrightarrow{AB},$ $\overrightarrow{AD},$ and $\overrightarrow{AE},$ as described above where $A$ is at the origin. $\overrightarrow{AD} = \mathbf{w}$, $\overrightarrow{AB} = 2\mathbf{v}$, and $\overrightarrow{AE} = \mathbf{u}$. Compute
\[
\frac{BH^2 + CG^2 + DF^2 + AE^2}{AB^2 + AD^2 + AE^2}.
\]

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