
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 that $P$ is a point on the semicircle $C$: $\begin{cases} x=\cos \theta, \\ y=\sin \theta \end{cases}$ (where $\theta$ is the parameter, $0\leqslant \theta\leqslant \pi$), the coordinates of point $A$ are $(1,0)$, $O$ is the origin, and point $M$ is on the ray $OP$. The length of segment $OM$ and the arc $\overline{AP}$ of $C$ are both $\dfrac{\pi}{3}$.

$(1)$ Establish a polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, and find the polar coordinates of point $M$;

$(2)$ Find the parametric equation of line $AM$.

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