
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 $xOy$, a polar coordinate system is established with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of curve $C_{1}$ is $\rho =4\sin \theta$. If $P$ is a moving point on curve $C_{1}$, rotating the line segment $OP$ clockwise around point $O$ by $60^{\circ}$ results in $OQ$, and the trajectory of point $Q$ is curve $C_{2}$.  
$(1)$ Find the polar coordinate equation of curve $C_{2}$.  
$(2)$ In the polar coordinate system, point $M(4,\frac{{2\pi}}{3})$, the ray $θ=\frac{\pi}{6}$ ($\rho\geq0$), intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively, which are distinct from the pole $O$. Find the area of $\triangle MAB$.

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