
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 faces of a cube with side length $10$. A beam of light emanates from vertex $A$ and reflects off face $EFGH$ at point $Q$, which is $4$ units from $\overline{EH}$ and $3$ units from $\overline{EF}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $p\sqrt{q}$, where $p$ and $q$ are integers and $q$ is not divisible by the square of any prime. Find $p+q$.

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