
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
Given in the Cartesian coordinate system, the parametric equation of curve $C$ is $\begin{cases}x=1+ \sqrt{5}\cos a \\ y=2+ \sqrt{5}\sin a\end{cases} (a$ is the parameter$)$, line $l_{1}:x=0$, line $l_{2}:x-y=0$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis $(\text{using the same unit of length})$ to establish the polar coordinate system.

$(1)$ Find the polar equations of curve $C$ and lines $l_{1}, l_{2}$;

$(2)$ If line $l_{1}$ intersects curve $C$ at points $O$ and $A$, and line $l_{2}$ intersects curve $C$ at points $O$ and $B$, find the length of segment $AB$.

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