
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
In order to revitalize the rural economy and increase the income of villagers, a village committee leader led the villagers to promote agricultural products through live streaming online. In the trial sales of $30$ days, the selling price $p$ (in yuan per kilogram) on the $x$-th day $(1\leqslant x\leqslant 30$ and $x$ is an integer) is related to $x$ by the function $p=\left\{{\begin{array}{l}{mx+n,}&{1≤x＜20, \text{ where } x \text{ is an integer}}\\{30,}&{20≤x≤30, \text{ where } x \text{ is an integer}}\end{array}}\right.$. The sales volume $q$ (in kilograms) on the $x$-th day is related to $x$ by the function $q=x+10$. It is known that the selling price on the $5$th day is $50$ yuan per kilogram, and on the $10$th day is $40$ yuan per kilogram. Let the sales revenue on the $x$-th day be $W$ yuan. Find:<br/>$(1) m=$______, $n=$______; <br/>$(2)$ the function relationship between the sales revenue $W$ in yuan and $x$; <br/>$(3)$ how many days in the 30-day trial sales period had sales revenue exceeding 1000 yuan?

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