
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
(Selective 4-4: Coordinate Systems and Parametric Equations)

In the rectangular coordinate system $xOy$, there is a circle ${C_1}:{(x - 2)^2} + {(y - 4)^2} = 20$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. (${C_2}:\theta = \frac{\pi }{3}(\rho \in R)$)

(1) Find the polar coordinate equation of ${C_1}$ and the rectangular coordinate equation of ${C_2}$;

(2) If the polar coordinate equation of the line ${C_3}$ is $\theta = \frac{\pi }{6}(\rho \in R)$, and the intersection points of ${C_2}$ and ${C_1}$ are $O$ and $M$, and the intersection points of ${C_3}$ and ${C_1}$ are $O$ and $N$, find the area of $\Delta OMN$.

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