
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 ex-radii of a triangle are  $10\frac{1}{2}, 12$  and  $14$ . If the sides of the triangle are the roots of the cubic  $x^3-px^2+qx-r=0$ , where  $p, q,r $  are integers , find the nearest integer to   $\sqrt{p+q+r}.$ 

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