
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 scientific research institution has developed a certain high-tech product, which has now entered the experimental stage. It is known that the startup capital for the experiment is 100,000 yuan. Starting from the first day of the experiment and continuing every day, the experimental cost on the $x$th day is $(px+280)$ yuan ($x\in\mathbb{N}^*$), and the total experimental cost for 30 days is 17,700 yuan.
(1) Find the value of $p$ and the number of days of the experiment when the average daily cost is minimized;
(2) A well-known enterprise sponsors the experiment for $x$ days with a total of $(-qx^2+50000)$ yuan ($q>0$). To ensure product quality, the experiment must last at least 50 days. If the experiment is to end when the average daily actual cost is minimized, find the range of values for $q$. (Actual cost = startup capital + experimental cost - sponsorship)

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