
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
It is known that P is a moving point on the circle A: $(x+1)^2 + y^2 = 12$, and point B is at $(1,0)$. The perpendicular bisector of segment PB intersects the radius PA at point T. Let the locus of point T be $\Gamma$.
(I) Find the equation of the curve $\Gamma$;
(II) Suppose M and N are two moving points on $\Gamma$ and the midpoint H of MN lies on the circle $x^2 + y^2 = 1$. Find the minimum distance from the origin to the line segment MN.

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