
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 _\(x\)_ as the polar axis, a polar coordinate system is established. It is known that the curve _\(C\)_ is defined by _\(\rho\sin^2\theta=2a\cos\theta\)_ \((a > 0)\), and the line _\(l\)_ passing through point _\(P(-2,-4)\)_ has the parametric equation: \(\begin{cases}x=-2+ \frac{\sqrt{2}}{2}t \\ y=-4+ \frac{\sqrt{2}}{2}t\end{cases}\) (where _\(t\)_ is the parameter), and the line _\(l\)_ intersects the curve _\(C\)_ at points _\(M\)_ and _\(N\)_.

(1) Write the Cartesian coordinate equation of curve _\(C\)_ and the general equation of line _\(l\)_.

(2) If \(|PM|\), \(|MN|\), and \(|PN|\) form a geometric sequence, find the value of _\(a\)_.

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