
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 certain online store monitored and statistically analyzed the sales price and sales volume of a seasonal product over the past $20$ days. The sales price $p$ (in yuan) on the $x$-th day ($1 \leqslant x \leqslant 20, x \in \mathbb{N}$) is given by $p= \begin{cases} 44+x,1 \leqslant x \leqslant 6 \\ 56-x,6 < x \leqslant 20 \end{cases}$, and the sales volume $q$ on the $x$-th day is given by $q= \begin{cases} 48-x,1 \leqslant x \leqslant 8 \\ 32+x,8 < x \leqslant 20 \end{cases}$. The cost of the product is known to be $25$ yuan per unit.

(I) Write the sales revenue $t$ as a function of the $x$-th day;
(II) Calculate the profit on the $7$-th day;
(III) On which day was the profit maximum? Calculate the maximum profit.

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