
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 $\theta$, given the curve $C_{1}$: $\left\{\begin{array}{l}{x=2+2\cos\theta}\\{y=2\sin\theta}\end{array}\right.$ ($\theta$ is the parameter, $\theta \in [0,\pi]$), in the polar coordinate system, the curve $C_{2}$ is a circle with center at $({1,\frac{\pi}{2}})$ and passing through the pole $O$. <br/>$(1)$ Write down the general equation of curve $C_{1}$ and the polar coordinate equation of curve $C_{2}$ respectively; <br/>$(2)$ The line $l: \theta =\frac{\pi}{4}$ ($\rho \in R$) intersects the curves $C_{1}$ and $C_{2}$ at points $M$ and $N$ (distinct from the pole $O$), find $|MN|$.

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