
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
The vertex of the parabola $C$ is at the origin $O$, and the focus is on the $x$-axis. The line $l: x=1$ intersects $C$ at points $P$ and $Q$, where $OP \perp OQ$. Point $M(2,0)$ is given, and circle $M$ is tangent to line $l$.  
$(1)$ Find the equations of $C$ and circle $M$.  
$(2)$ Let $A_{1}$, $A_{2}$, $A_{3}$ be three points on $C$, and lines $A_{1}A_{2}$, $A_{1}A_{3}$ are both tangent to circle $M$. Determine the position relationship between line $A_{2}A_{3}$ and circle $M$, and explain the reason.

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