
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 chain store sells a certain brand of goods, with a cost price of $60$ yuan per item. As it is a new store opening, the chain store decided to conduct a trial operation for the first $10$ days of the month. During this period, the selling price of the goods is $80$ yuan per item. According to research, the relationship between the number of items sold on a given day $y_{1}$ (in pieces) and the day $x$ is given by $y_{1}=x^{2}+bx+c$ for $1\leqslant x\leqslant 10$. It is known that the number of items sold on the 4th day is 40 pieces, and on the 6th day is 44 pieces. After the promotion, the chain store redefined the selling price of the goods to $100$ yuan per item. The number of items sold per day also changed: the relationship between the number of items sold on a given day $y_{2}$ (in pieces) and the day $x$ is $y_{2}=2x+8$ for $11\leqslant x\leqslant 31$.
  
$(1)$ Find the functional relationship of $y_{1}$ with respect to $x$;  
$(2)$ If the daily gross profit on a certain day is $1120$ yuan, find the value of $x$;  
$(3)$ Because the chain store is a new store opening, after the trial operation, the manufacturer provides corresponding preferential policies to the chain store: when the daily sales volume of this chain store reaches 60 items (excluding 60), for each additional sale of 1 item, the cost price of all products sold on that day is reduced by 2 yuan. Let the total gross profit when the store sells more than 60 items in a day be $W$. Write directly the analytical expression of $W$ with respect to $x$, and the range of the independent variable $x$: ______.

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