
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 market survey shows that the daily sales volume and daily sales price of a certain product in the past 20 days are functions of the sales time $t$ (days). The daily sales volume $f(t)$ (in units) approximately satisfies $f(t)=-t+30$ ($1\leqslant t\leqslant 20$, $t \in \mathbb{N}^*$), and the daily sales price $g(t)$ (in yuan) approximately satisfies $g(t)= \begin{cases} 2t+40, & 1\leqslant t\leqslant 10, t \in \mathbb{N}^* \\ 15, & 11\leqslant t\leqslant 20, t \in \mathbb{N}^* \end{cases}$.

(1) Write the function relationship $S$ of the daily sales amount of the product with respect to time $t$;
(2) When does $t$ make the daily sales amount $S$ reach its maximum? And find the maximum value.

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