
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 plane Cartesian coordinate system $xOy$, a line $l$ passes through $M(2,1)$ with an inclination angle of $\dfrac{\pi}{4}$. Using the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. Both coordinate systems use the same unit of length. The polar equation of circle $C$ is $\rho=4 \sqrt{2}\sin \left(\theta+ \dfrac{\pi}{4}\right)$.

$(1)$ Find the parametric equation of line $l$ and the Cartesian equation of circle $C$;

$(2)$ Suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $\dfrac{1}{|MA|}+ \dfrac{1}{|MB|}$.

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