
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 \(EFGH\) and \(ABFE\) be two faces of a rectangular prism with dimensions \(AB = EF = 13,\ BC = FG = 15,\) and \(AE = EH = 10\). A beam of light emanates from vertex \(E\) and reflects off face \(ABFE\) at point \(Q\), which is 9 units from \(\overline{AB}\) and 6 units from \(\overline{AE}\). Calculate the length of the light path from the time it leaves point \(E\) until it first returns to any vertex of the prism, in the simplest radical form \(p\sqrt{q}\), where \(p\) and \(q\) are integers and \(q\) is not divisible by the square of any prime. Determine \(p+q\).

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