
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 two circles C<sub>1</sub>: (x+1)<sup>2</sup>+y<sup>2</sup> = 1 and C<sub>2</sub>: (x-4)<sup>2</sup>+y<sup>2</sup> = 4.
(1) Draw a line l through the center of C<sub>1</sub> at an angle θ to intersect circle C<sub>2</sub> at points A and B, where A is the midpoint of C<sub>1</sub>B. Find sinθ.
(2) Through point P(m, 1), draw two secant lines l<sub>1</sub> and l<sub>2</sub> to circle C<sub>2</sub>, with the midpoints of the corresponding chords being M and N. Does the circle through points P, M, N, and C<sub>2</sub> always pass through a fixed point (different from point C<sub>2</sub>)? If it passes through a fixed point, find that point; otherwise, explain why it does not.
(3) Draw two tangents to circle C<sub>1</sub> from a point Q(x<sub>0</sub>, y<sub>0</sub>) on circle C<sub>2</sub>, and suppose the tangents intersect the y-axis at points S and T. Determine the range of lengths for the line segment ST.

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