
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 the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{2}$, and passing through the point $A(\frac{3}{2}, -\frac{1}{2})$.
(1) Find the equation of the ellipse;
(2) Find the coordinates of a point $P$ on the ellipse $C$ such that its distance to the line $l$: $x+y+4=0$ is minimized, and find this minimum distance.

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