
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 parametric equations of the line $l$ are $\left\{\begin{array}{l}x=3-\frac{{\sqrt{2}}}{2}t\\ y=1+\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is a parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of the curve $C$ is $\rho =4\cos \theta +6\sin \theta $.  
$(1)$ Find the Cartesian equation of the curve $C$.  
$(2)$ Let the curve $C$ intersect the line $l$ at points $M$ and $N$, and the coordinates of point $A$ are $\left(3,1\right)$. Find $|AM|+|AN|$.

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