
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
In a triangle, the angles are in arithmetic progression, with the smallest angle being $30^\circ$. The side lengths of the triangle are $6, 8, x$. Compute the sum of the possible values of $x$ given that it can be expressed as $p + \sqrt{q} + \sqrt{r}$, where $p, q, r$ are integers.
**A)** $7 + \sqrt{35} + \sqrt{83}$  
**B)** $6 + \sqrt{36} + \sqrt{83}$  
**C)** $7 + \sqrt{36} + \sqrt{84}$  
**D)** $7 + \sqrt{36} + \sqrt{83}$

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