
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 curve $C$ are $\left\{{\begin{array}{l}{x=2\cos\alpha,}\\{y=3\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of the line $l$ is $2\rho \cos \theta +\rho \sin \theta -6=0$.  
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;  
$(2)$ Through any point $P$ on curve $C$, draw a line that makes a $45^\circ$ angle with line $l$ and intersects $l$ at point $A$. Find the minimum value of $|PA|$.

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