
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 $C$ is tangent to the $y$-axis at point $A(0,1)$, and the chord intercepted by the $x$-axis has a length of $2\sqrt{3}$, with the center of circle $C$ in the first quadrant.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If point $P$ is a moving point on line $l: 3x+4y+5=0$, and a tangent to circle $C$ is drawn from point $P$, with the point of tangency at $B$, find the minimum area of $\triangle PBC$.

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