
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 $EFGH$ be two faces of a cube with $AB=10$. A beam of light is emitted from vertex $A$ and reflects off face $EFGH$ at point $Q$, which is 6 units from $\overline{EH}$ and 4 units from $\overline{EF}$. The beam is then reflected off other faces of the cube. Determine the length of the light path from its emission from $A$ until it reaches another vertex of the cube for the first time, expressed in the form $s\sqrt{t}$, where $s$ and $t$ are integers with $t$ having no square factors. Provide $s+t$.

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