
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 circle $C$ passes through points $A(3,3)$ and $B(2,4)$, and the center of the circle $C$ is on the line $y=3x-5$.  
$(1)$ Find the standard equation of circle $C$;  
$(2)$ Let $P(-m,0)$ and $Q(m,0)$ $(m > 0)$, if there exists a point $M$ on circle $C$ such that point $M$ is also on the circle with diameter $PQ$, find the range of the real number $m$.

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