
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 the line $C_{1}$ are $\left\{\begin{array}{l}x=t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ ($t$ is the parameter, $0＜\alpha＜\frac{π}{2}$). $C_{1}$ is rotated counterclockwise around the origin by $\frac{π}{2}$ to obtain $C_{2}$. Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, using the same unit length.<br/>$(1)$ Write down the polar equations of $C_{1}$ and $C_{2}$;<br/>$(2)$ If the polar equation of curve $C_{3}$ is $\rho =8\sin \theta $, and $C_{1}$ intersects $C_{3}$ at point $A$, and $C_{2}$ intersects $C_{3}$ at point $B$ (where $A$ and $B$ are not coincident with point $O$), find the maximum area of $\triangle AOB$.

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