
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 2017, under the guidance of the national innovation-driven strategy, the Beidou system, as a national high-tech project and an open innovation platform, had more than 1400 Beidou base stations spread across the country. Tens of thousands of sets of equipment formed a "net" between the stars and the earth. The domestic positioning accuracy reached the sub-meter level, and in some areas, it reached the decimeter level, with the highest accuracy even reaching the centimeter or millimeter level. Recently, the Beidou-3 project spent 90,000 yuan to build a small device. It is known that from the first day of use, the daily maintenance cost of this device on the $n$-th day is $\frac{n}{2}+99.5$ yuan ($n\in\mathbb{N}^*$). To use it until it is "most economical to scrap" (meaning that the average daily cost of using this device is the lowest), how many days in total was it used, and what was the average daily cost?

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