
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, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=-\sqrt{3}t}\\{y=t}\end{array}}\right.$ ($t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{{\begin{array}{l}{x=4\cos\theta}\\{y=4\sin\theta}\end{array}}\right.$ ($\theta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of $C_{1}$ and $C_{2}$;<br/>$(2)$ Let the coordinates of point $P$ be $\left(1,0\right)$, a line $l$ passes through point $P$, intersects $C_{2}$ at points $A$ and $B$, and intersects $C_{1}$ at point $M$. Find the maximum value of $\frac{{|{PA}|⋅|{PB}|}}{{|{PM}|}}$.

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