
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
Bike-sharing brings many conveniences to the public transportation. A company purchased a batch of bicycles for public use in a certain area. According to market analysis, the cumulative profit $y$ (unit: Yuan) of each bicycle from operation is related to the number of operating days $x$ ($x \in \mathbb{N}^*$) by the function $y = -\frac{1}{2}x^2 + 60x - 800$.
$(1)$ To ensure the cumulative profit from operation is more than $800$ Yuan, find the range of operating days.
$(2)$ How many days should each bicycle operate to maximize the average daily operating profit $\frac{y}{x}$?

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