
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
(1) Given an ellipse C: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with an eccentricity of $\frac{1}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse, which is tangent to the line $\sqrt{7}x - \sqrt{5}y + 12 = 0$. Find the equation of ellipse C;  
(2) Given $\bigcirc A_{1}: (x+2)^{2} + y^{2} = 12$ and point $A_{2}(2, 0)$, find the equation of the trajectory of the center $P$ of a moving circle that passes through point $A_{2}$ and is tangent to $\bigcirc A_{1}$.

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