
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 supermarket plans to sell a certain brand of mooncakes for $50$ days before the Mid-Autumn Festival, with a purchase price of $18$ yuan per kilogram. Let the selling price on day $x$ be $y$ (yuan per kilogram), and the sales volume be $m$ (kilograms). Based on past sales experience, the supermarket has derived the following sales rules: ① When $1\leqslant x\leqslant 30$, $y=40$; when $31\leqslant x\leqslant 50$, $y$ and $x$ satisfy a linear function relationship, and when $x=36$, $y=37$; when $x=44$, $y=33$. ② The relationship between $m$ and $x$ is $m=5x+50$. <br/>$(1)$ The relationship between $y$ and $x$ when $31\leqslant x\leqslant 50$ is ______; <br/>$(2)$ For what value of $x$ is the daily sales profit $W$ (yuan) maximized? What is the maximum profit? <br/>$(3)$ If the supermarket wants the daily sales profit $W$ from the $31$st to the $35$th day to increase with the increase of $x$, then it needs to increase the selling price by $a$ yuan per kilogram. Find the range of values for $a$.

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