
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
Following a market survey, it was found that the daily sales volume and daily sales price of a certain product over the past 20 days were functions of the sales time $t$ (days). The daily sales volume (unit: items) approximately satisfies the function $f(t) = -t + 30$ ($1 \leq t \leq 20$, $t \in \mathbb{N}^*$), and the daily sales price (unit: yuan) approximately satisfies the function $g(t) = \begin{cases} 2t + 40, & 1 \leq t \leq 10, t \in \mathbb{N}^* \\ 15, & 11 \leq t \leq 20, t \in \mathbb{N}^* \end{cases}$.
(1) Write out the function relationship of the daily sales revenue $S$ with respect to time $t$.
(2) At what value of $t$ is the daily sales revenue $S$ maximum? Also, calculate the maximum value.

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