
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 $C: x^2+y^2+ax-4y+1=0$ ($a\in \mathbb{R}$), a line with a slope of 1 passing through a fixed point $P(0,1)$ intersects circle $C$ at points $A$ and $B$, with $P$ being the midpoint of segment $AB$.
- (I) Find the value of $a$;
- (II) Let $E$ be a point on circle $C$ different from $A$ and $B$. Find the maximum area of $\triangle ABE$;
- (III) From a point $M$ outside the circle $C$, draw a tangent line to circle $C$ at point $N$, and let $|MN|=|MP|$. Find the minimum value of $|MN|$ and the coordinates of point $M$ when $|MN|$ is at its minimum.

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