
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 $xoy$, the curve $C: (x-1)^2 + y^2 = 1$ is given. A line $l$ passes through the point $P(m,0)$ and has an inclination angle of $\frac{\pi}{6}$. Taking the origin $O$ of the Cartesian coordinate system $xoy$ as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit length.

$(1)$ Write the polar equation of curve $C$ and the parametric equation of line $l$.

$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and $|PA| \cdot |PB| = 1$, find the value of the real number $m$.

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