
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
The angles in a triangle are in geometric progression, and the side lengths are $6, 7, x$. The sum of the angles doubles the smallest angle. Express the sum of the possible values of $x$ in the form $p+\sqrt{q}+\sqrt{r}$, where $p, q$, and $r$ are integers, and determine the value of $p+q+r$.

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