
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, a line $l$ with an inclination angle of $\alpha$ passes through point $M(-2,-4)$. Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho \sin^2\theta = 2\cos\theta$.

$(1)$ Write the parametric equation of line $l$ ($\alpha$ is a constant) and the Cartesian equation of curve $C$;

$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and $|MA|\cdot |MB|=40$, find the value of the inclination angle $\alpha$.

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