
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 polar coordinate system, the polar equation of circle $C$ is $ρ=2\sqrt{2}sin({θ+\frac{π}{4}})$, and the polar equation of line $l$ is $ρsin({θ+\frac{π}{4}})=4$. Establish a rectangular coordinate system $xOy$ with the pole as the origin and the polar axis as the positive x-axis.<br/>$(1)$ Find the rectangular coordinate equations of circle $C$ and line $l$;<br/>$(2)$ If the ray $\theta =\alpha \left(\rho\ \ \gt 0\right)$ intersects circle $C$ and line $l$ at points $P$ and $Q$, where $α∈({0,\frac{π}{2}})$, find the maximum value of $\frac{{|{OP}|}}{{|{OQ}|}}$.

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