
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 $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=\cos\alpha}\\{y=1+\sin\alpha}\end{array}\right.$ ($\alpha$ is the parameter). Point $M$ is a moving point on $C_{1}$, and point $P$ satisfies $\overrightarrow{OP}=3\overrightarrow{OM}$. The locus of point $P$ is curve $C_{2}$.  
$(Ⅰ)$ Find the parametric equations of $C_{2}$.  
$(Ⅱ)$ In the polar coordinate system with pole $O$ and the positive $x$-axis as the polar axis, the line $y=\frac{{\sqrt{3}}}{3}x$ intersects curve $C_{1}$ at point $A$ (other than the pole) and intersects curve $C_{2}$ at point $B$. After converting the equations of curves $C_{1}$ and $C_{2}$ into polar coordinate equations, find $|AB|$.

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