
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 circle $C$: $x^2 + y^2 - 4x - 14y + 45 = 0$ and a point $Q(-2, 3)$.
(1) If $P(A, A+1)$ lies on the circle, find the length of the line segment $PQ$ and the slope of the line $PQ$.
(2) If $M$ is any point on the circle, find the maximum and minimum values of $|MQ|$.

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