
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 with center $C$ that passes through points $A(1,1)$ and $B(2,-2)$, and the center $C$ is on the line $l$: $x+3y+3=0$.
(1) Find the equation of circle $C$.
(2) If $P$ is a moving point on the line $3x+4y-21=0$, and $PM$, $PN$ are the two tangents of circle $C$, with $M$ and $N$ being the points of tangency. Let $|PC|=t$, express the area $S$ of the quadrilateral $PMCN$ as a function of $t$, and find the minimum value of this function.

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