
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
A park plans to build a Ferris wheel, the perimeter of which is a circle with a circumference of $k$ meters. Seats are to be installed on this circle, and each seat is connected to the support at the center of the circle by a straight steel pipe. According to the budget, the cost of each steel pipe connecting a seat to the support on the Ferris wheel is $8k$ yuan per piece, and when the arc length between two adjacent seats is $x$ meters, the total cost of the steel pipes between two adjacent seats and one of the seats is $\left[ \frac {(1024 \sqrt {x}+20)x}{100}+2\right]k$ yuan. Assuming the seats are evenly distributed, with at least two seats, all seats are considered as points, and other factors are not considered, let the total cost of the Ferris wheel be $y$ yuan.  
(1) Try to write the function relationship of $y$ with respect to $x$, and state its domain;  
(2) When $k=100$ meters, try to determine the number of seats to minimize the total cost?

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