
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 polar equation of curve \(C\) is \(ρ\sin^2θ=2a\cos θ\; (a > 0)\), and the parametric equation of line _\(l\)_ is \(\begin{cases}x=-2+ \frac{ \sqrt{2}}{2}t \\ y=-4+ \frac{ \sqrt{2}}{2}t\end{cases}\) (with \(t\) as the parameter). Line _\(l\)_ intersects curve \(C\) at points \(A\) and \(B\).  
\((1)\) Write the Cartesian coordinate equation of curve \(C\) and the general equation of line _\(l\)_;  
\((2)\) If \(|AB|=2 \sqrt{10}\), find the value of _\(a\)_.

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