
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
In the diagram, $\triangle A O B$ is a triangle with coordinates $O=(0,0), A=(0,30)$, and $B=(40,0)$. Let $C$ be the point on $A B$ for which $O C$ is perpendicular to $A B$.
(a) Determine the length of $O C$.
(b) Determine the coordinates of point $C$.
(c) Let $M$ be the center of the circle passing through $O, A$, and $B$. Determine the length of $C M$.

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