
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 that point $M$ lies on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.

(1) If $P(a,a+1)$ is on circle $C$, find the length of segment $PQ$ and the slope of line $PQ$;

(2) Find the maximum and minimum values of $|MQ|$;

(3) If $M(m,n)$, find the maximum and minimum values of $\frac{n-{3}}{m+{2}}$.

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