
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 $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho\cos (\theta- \frac{\pi}{3}) =1$. $M$ and $N$ are the intersections of curve $C$ with the $x$-axis and $y$-axis, respectively.  
$(1)$ Write the Cartesian coordinate equation of curve $C$, and find the polar coordinates of $M$ and $N$;  

$(2)$ Suppose the midpoint of $M$ and $N$ is $P$, find the polar equation of line $OP$.

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