
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 $xOy$, given the curve $C: x^{2}+y^{2}=|x|+y$ (where $y \gt 0$), the curve $C_{1}: \left\{\begin{array}{c}x=t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is a parameter, $t \gt 0$), and the curve $C_{2}: \left\{\begin{array}{c}x=-t\sin\alpha\\ y=t\cos\alpha\end{array}\right.$ (where $t \gt 0, 0 \lt \alpha \lt \frac{\pi}{2}$). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equation of $C$;<br/>$(2)$ If the curve $C$ intersects $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively, find the maximum area of $\triangle OAB$.

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