
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
Given that the equation of the line $l$ in rectangular coordinates is $x+y-2=0$, and the equation of the curve $C$ is $\left(x-2\right)^{2}+y^{2}=4$. Establish a polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of the line $l$ and the curve $C$;<br/>$(2)$ If the ray $θ=\frac{π}{4}({ρ≥0})$ intersects the line $l$ and the curve $C$ at points $M$ and $N$ respectively $(N$ is different from the origin $O)$, find $|MN|$.

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