
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 plane rectangular coordinate system $xOy$, given the curve $C$: $\left\{{\begin{array}{l}{x=\cos\alpha}\\{y=2\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter), and the line $l$: $\left\{{\begin{array}{l}{x=1+t}\\{y=3-2t}\end{array}}\right.$ ($t$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of the curve $C$ and the line $l$;<br/>$(2)$ Point $P$ lies on the line $l$, the ray $OP$ intersects the curve $C$ at point $R$, point $Q$ lies on the ray $OP$, and satisfies $5|OR|^{2}=4|OP||OQ|$. Find the rectangular coordinate equation of the locus of point $Q$.

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