
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
In a Cartesian coordinate system, establish a polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that the polar equation of curve C is $\rho \cos^2\theta = 2\sin\theta$. The tangent line at point $M(2\sqrt{2}, \frac{\pi}{4})$ is the line $l$.  
(1) Find the equation of the line $l$ in Cartesian coordinates.  
(2) Given that point $P$ lies on the ellipse $\frac{x^2}{3} + \frac{y^2}{4} = 1$, determine the range of distances from point $P$ to the line $l$.

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