
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 $ABCD$ and $EFGH$ be two adjacent faces of a cube, where $AB=10$ units. A beam of light is emitted from vertex $B$ and reflects off face $EFGH$ at a point $Q$, which is 3 units from $\overline{EH}$ and 6 units from $\overline{EF}$. The beam continues its motion reflecting off the faces of the cube. Determine the length of the light's path from the point it leaves vertex $B$ until it next reaches a vertex of the cube, and express your answer in the form $p\sqrt{q}$, where $p$ and $q$ are integers, and $q$ is not divisible by the square of any prime.

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