
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 $xoy$, the line $l$ passing through $M(2,1)$ has an inclination angle of $\frac{\pi}{4}$. The polar coordinate system is established with the origin as the pole and the positive $x$-axis as the initial line, using the same unit length as the Cartesian coordinate system. The polar equation of circle $C$ is $\rho=4\sqrt{2}\sin(\theta+\frac{\pi}{4})$.
$(1)$ 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 $\frac{1}{\left|MA\right|}+\frac{1}{\left|MB\right|}$.

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