
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
During a market survey, it was found that the sales volume (in items) and price (in yuan) of a certain product in the last 90 days are both functions of time t (in days), approximately satisfying $f(t)= $
$$
\begin{cases}
\frac{1}{4}t+10, & 1\leq t\leq 40, t\in\mathbb{N}^{+}\\
t-20, & 40<t\leq 90, t\in\mathbb{N}^{+}
\end{cases}
$$
and $g(t)= $
$$
\begin{cases}
-10t+630, & 1\leq t\leq 40, t\in\mathbb{N}^{+}\\
-\frac{1}{10}t^2+10t-10, & 40<t\leq 90, t\in\mathbb{N}^{+}
\end{cases}
$$
1. Write the function representing the product's daily sales revenue S (the product of the sales volume and the price) as a function of time t.
2. Calculate the maximum value of the product's daily sales revenue S.

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