
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 $ABCD$, $AEHD$, and $EFGH$ be three adjacent faces of a cube with side length $10$. A beam of light emanates from vertex $A$ and reflects off face $EFGH$ at point $P$, which is $4$ units from $\overline{EH}$ and $3$ units from $\overline{EF}$. Determine the length of the light path from when it leaves point $A$ until it reaches vertex $D$ directly for the first time.

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