
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 is trying to market a new type of product. The cost price is $32$ yuan per item, the trial marketing period is $25$ days, and the selling price $y$ (in yuan per item) is related to the number of days $x$ as follows: for $1\leqslant x\leqslant 14$, $y=k_{1}x+40$; for $15\leqslant x\leqslant 25$, $y=\frac{{k}_{2}}{x}+32$. During the trial marketing period, the sales volume is $p=-x+48$.  
$(1)$ When $x=4$, $y=42$; when $x=20$, $y=37$, find the values of $k_{1}$ and $k_{2}$;  
$(2)$ For $15\leqslant x\leqslant 25$, at which day does the online store have the maximum sales profit $W$?  
$(3)$ In the first 14 days of actual sales, the government decides to subsidize the company with $a$ yuan (where $a>1$) for each item sold. The company finds that with the subsidy, the daily sales profit in the first 14 days increases with the number of days $x$. Find the range of values for $a$.

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