
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 an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \, (a>b>0)$$ with focal length of $2\sqrt{2}$, and it passes through the point $A\left(\frac{3}{2}, -\frac{1}{2}\right)$.
(1) Find the equation of the ellipse;
(2) On ellipse C, find the point P such that the distance from P to the line $l: x+y+4=0$ is the shortest; find the coordinates of P and calculate the shortest distance.

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