
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 line $l$ intersects circle $C$: $x^2+y^2+2x-4y+a=0$ at points $A$ and $B$, and the midpoint of chord $AB$ is $P(0,1)$.
(I) If the radius of circle $C$ is $\sqrt{3}$, find the value of the real number $a$;
(II) If the length of chord $AB$ is $6$, find the value of the real number $a$;
(III) When $a=1$, circles $O$: $x^2+y^2=4$ and $C$ intersect at points $M$ and $N$, find the length of the common chord $MN$.

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