
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 $M$ is a moving point on the circle $C: \left(x-4\right)^{2}+y^{2}=4$, and point $N$ is any point on the line $y=x$ except for the origin $O$, find the maximum value of the projection of vector $\overrightarrow{OM}$ onto vector $\overrightarrow{ON}$.

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