
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 point $P$ is on the semicircle $C$: $\begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases}$ ($\theta$ is the parameter, $0\leqslant \theta\leqslant \pi$), the coordinates of point $A$ are $(1,0)$, $O$ is the coordinate origin, and point $M$ is on the ray $OP$. The length of the line segment $OM$ and the arc $\widehat{AP}$ are both $\frac{\pi}{3}$.
(1) Establish a polar coordinate system with $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Find the polar coordinates of point $M$.
(2) Find the parametric equation of the line $AM$.

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