
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 the parametric equation of curve $C$: 
$$
\begin{cases} 
x=\cos \alpha, \\
y=1+\sin \alpha 
\end{cases}
$$
where $\alpha$ is the parameter, and $\alpha \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.

$(1)$ Find the polar equation of curve $C$.

$(2)$ Let $T$ be a point on curve $C$, and let the length of the chord cut by line $OT$ on curve $C$ be $\sqrt{3}$. Find the polar coordinates of point $T$.

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