
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  $AB=12.$   A beam of light emanates from vertex  $A$  and reflects off face  $BCFG$  at point  $P,$  which is 7 units from  $\overline{BG}$  and 5 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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