
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 is emitted 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 its path, reflecting off the faces of the cube. The length of the light path from when it leaves point $A$ until it reaches another vertex of the cube for the first time is expressed as $r\sqrt{s}$, where $r$ and $s$ are integers and $s$ is not divisible by the square of any prime. Determine $r+s$.

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