
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 \(BCFG\) be two faces of a cube with \(AB=10\). A beam of light emanates from vertex \(A\) and reflects off face \(BCFG\) at point \(P\), which is 3 units from \( \overline{BG} \) and 4 units from \( \overline{BC} \). 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 \( m\sqrt{n} \), where \( m \) and \( n \) are integers and \( n \) is not divisible by the square of any prime. Find \( m+n \).

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