
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 $F_1$ and $F_2$ as the left and right foci of the ellipse $C$ defined by $\frac{x^2}{4} + \frac{y^2}{3} = 1$, the circle with diameter $F_1F_2$ is symmetrical to the circle $C$ with respect to the line $x+y-2=0$.

(1) Find the equation of circle $C$.

(2) Through the point $P(m,0)$, draw the tangent to circle $C$ and find the minimum value of the tangent's length as well as the coordinates of the corresponding point $P$.

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