
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
1. There are 1000 students participating in a math competition, numbered from 0001 to 1000. A sample of 50 students is selected using systematic sampling and divided into 50 parts. The first part consists of students numbered 0001 to 0020, and a random draw from this part results in student number 0015. What is the number of the third student drawn?
2. An ellipse C has its center at the origin, with its major axis on the x-axis, and $c = \frac{\sqrt{3}}{2}a$. The sum of the distances from a point on the ellipse to its two foci is 12. What is the equation of ellipse C?
3. A moving point M(x, y) is such that its distance to a fixed point (2, 0) is one unit less than its distance to the line x = -3. What is the equation of the path traced by moving point M?
4. Given points A(1, 0) and B(3, 0), find the range of values of k such that there is a point P on the line y = kx + 1 that is perpendicular to segment AB.

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