
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
The price per unit $P($€$)$ of a certain product in the recent $30$ days and the relationship between time $t($day$)$ can be approximately expressed as $P= \begin{cases} -t+100,25\leqslant t\leqslant 30,t\in N \ \overset{t+20,1\leqslant t\leqslant 24,t\in N}{} \end{cases}$, and the daily sales volume $Q($unit$)$ and time $t($day$)$ The relationship between them can be approximately expressed as $Q=-t+40(1\leqslant t\leqslant 30,t∈N)$.
(1) Find the relationship between the daily sales $y$ of this product and time $t$;
(2) Find the maximum value of $y$, and indicate which day of the $30$ days has the largest daily sales.

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