
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 bike-sharing company has purchased a batch of bicycles for use by urban residents. According to market analysis, the cumulative profit per bike, $y$ (in yuan), and the number of operating days, $x$ ($x\in {N}^{*}$), satisfy the functional relationship $y=-\frac{1}{2}{x}^{2}+60x-800$.

(1) To make the cumulative profit per bike higher than 800 yuan, find the range of operating days.

(2) How many days should each bike be operated to maximize the average daily operating profit, $\frac{y}{x}$?

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