
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
After a market survey, the sales price (unit: Yuan) of a certain product over the past 50 days has been a function of the sales time $t$ (days), and the sales volume (unit: pieces) approximately satisfies $f(t)=-2t+200$ $(1 \leqslant t \leqslant 50, t \in \mathbb{N})$. The price (unit: Yuan) for the first 30 days is $g(t)= \frac{1}{2}t+30$ $(1 \leqslant t \leqslant 30, t \in \mathbb{N})$, and for the last 20 days is $g(t)=40$ $(31 \leqslant t \leqslant 50, t \in \mathbb{N})$.
- (I) Write the function relationship between the daily sales revenue $S$ (Yuan) and time $t$ (days);
- (II) Find the maximum value of the daily sales revenue $S$.

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