
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 plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}x=-1+2\cos\theta\\ y=1+2\sin\theta\end{array}\right.$ (where $\theta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis. The polar coordinate equation of the line $l$ is $\beta =\alpha (\alpha \in [0,\pi), \rho \in \mathbb{R})$. It is known that the line $l$ intersects the curve $C$ at points $M$ and $N$. <br/>$(1)$ Find the polar coordinate equation of curve $C$;<br/>$(2)$ Let the midpoint of line segment $MN$ be $P$. If $|OP|\leqslant \lambda$ always holds, find the range of real number $\lambda$.

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