
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 that passes through the origin and its center is on the ray $y=2x$ ($x>0$), with a radius of $\sqrt{5}$.
(1) Find the equation of circle C;
(2) If M is a moving point on the line $m: x+2y+5=0$, and N is a moving point on circle C, find the minimum value of $|MN|$ and the coordinates of point M when $|MN|$ reaches its minimum value.

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