
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
Given a line $y = \frac{\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\sqrt{3}$.

(1) Given a point $P(-1, \sqrt{7})$, and $Q$ is any point on circle $C$, find the maximum value of $|PQ|$.

(2) If a ray is drawn from the center of the circle to intersect circle $C$ at point $M$, find the probability that point $M$ lies on the minor arc $\hat{AB}$.

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