
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 $\triangle ABC$, the perimeter is 20, the area is $10 \sqrt{3}$, and $2 \angle A = \angle B + \angle C$.

1. Find the length of $BC$.
2. If point $P$ is on the incircle $\odot I$ of $\triangle ABC$ and $\overrightarrow{AI} = x \overrightarrow{AB} + y \overrightarrow{AC}$:
   1. Find the value of $x + y$.
   2. Find the range of $|\overrightarrow{AP} + \overrightarrow{BP} + \overrightarrow{CP}|$.

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