
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 is tangent to two circles $x^2+(y+4)^2=1$ and $x^2+(y-2)^2=1$, let the equation of the locus of the center of circle C be L. Suppose the minimum distance from a point on L to a point M $(x, y)$ is $m$, and the distance from point F $(0, 1)$ to point M $(x, y)$ is $n$.
(Ⅰ) Find the equation of the locus L of the center of circle C;
(Ⅱ) Find the equation of the locus Q of point M satisfying the condition $m=n$;
(Ⅲ) Investigate whether there exists a point B $(x_1, y_1)$ on the locus Q such that the area of the triangle formed by the tangent line at point B and the two coordinate axes is $\frac{1}{2}$. If it exists, find the coordinates of point B; if not, explain why.

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