
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 the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a coordinate system is established. It is known that curve C: $\rho\sin^2\theta=2a\cos\theta$ ($a>0$), and a line $l$ passing through point P(-2, -4) with a slope angle of $\frac{\pi}{4}$ intersects curve C at points M and N.  
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;  
(2) If $|PM|$, $|MN|$, and $|PN|$ form a geometric progression, find the value of $a$.

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