
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
In the Cartesian coordinate system xOy, the parabola is given by y = ax<sup>2</sup> - 4ax - 5a (a ≠ 0). <br /> (1) The parabola intersects the x-axis at points A and B (point A is to the left of point B). Find the coordinates of points A and B; <br /> (2) If point P (m, n) is a point on the parabola, under the condition a > 0, when m ≥ 0, the range of n is n ≥ -9. Find the analytical expression of the parabola; <br /> (3) When a = 1, the parabola y = ax<sup>2</sup> - 4ax - 5a is shifted upwards by m (m > 0) units to obtain a new parabola G. Let the abscissa of the intersection point of the new parabola G with the x-axis be t, and t satisfies $-\frac{1}{2}$ < t < $\frac{5}{2}$. Please directly write down the range of m.

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