
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 the triangle shown, for $\angle A$ to be the largest angle of the triangle, the side lengths are now given as $x+11$ for $BC$, $x+6$ for $AC$, and $3x+2$ for $AB$. Find the range of $x$ such that it satisfies these conditions and the triangle inequalities. What is the least possible value of $n-m$, expressed as a common fraction, for the interval $(m, n)$ in which $x$ lies?

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