
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 a line $l$: $(k-1)x-2y+5-3k=0$ ($k\in \mathbb{R}$) always passes through a fixed point $P$, and a circle $C$ passes through point $A(4,0)$ and point $P$, and the center of the circle is on the line $x-2y+1=0$.
$(1)$ Find the coordinates of the fixed point $P$;
$(2)$ Find the equation of circle $C$;
$(3)$ Given that point $P$ is one endpoint of the diameter of circle $C$, and the other endpoint is point $Q$. Is there a point $M(0,m)$ on the $y$-axis such that $\triangle PMQ$ is a right-angled triangle? If it exists, find the value of $m$; if not, explain why.

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