
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 $C$: $(x-0)^2+(y-3)^2=4$, a line $m$: $x+3y+6=0$, a moving line $l$ passing through point $A(1,0)$ intersects line $m$ at point $N$ and circle $C$ at points $P$ and $Q$. $M$ is the midpoint of $PQ$.

(1) Find the equation of line $l$ when $|PQ|=2\sqrt{3}$.

(2) Let $t=\overrightarrow{AM}\cdot\overrightarrow{AN}$. Determine whether $t$ is a constant value. If it is, find the value of $t$; if not, explain the reason.

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