
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 rectangular prism where $AB = 9$, $BC = 9$, and $AD = 18$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 8 units from $\overline{BG}$ and 6 units from $\overline{BC}$. Determine the length of the light path from when it leaves point $A$ until it reaches a vertex of the prism again, and express this in the form $m\sqrt{n}$ where $m$ and $n$ are integers, with $n$ not divisible by the square of any prime.

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