
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 research group of grade 10 students conducted a market survey on a certain daily-use product. The sales volume and price of this product in the past $100$ days are both functions of time $t$. The sales volume can be approximated as $g(t)=-t+110$ $(1 \leqslant t \leqslant 100, t \in N)$, the price for the first $40$ days was $f(t)=t+8$ $(1 \leqslant t \leqslant 40, t \in N)$, and the price for the last $60$ days was $f(t)=-0.5t+69$ $(41 \leqslant t \leqslant 100, t \in N)$.

1. Derive the relationship between the daily sales $S$ and time $t$.
2. Determine whether there exists a day with the highest sales in the past $100$ days, and if so, which day is it? (Note: Sales amount = Sales volume * Sales price)

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