
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 lanterns in the 50 days before the Spring Festival, with a cost price of $18$ yuan per lantern. Let the selling price of the $x$th day be $y$ (yuan per lantern) and the quantity sold be $n$ (lanterns). Based on past sales experience, the supermarket has derived the following sales rules:<br/>① The relationship between $y$ and $x$ is $y=-\frac{1}{2}x+55$;<br/>② The relationship between $n$ and $x$ is $n=5x+50$.<br/>$(1)$ Find the daily sales profit on the 10th day;<br/>$(2)$ When $34\leqslant x\leqslant 50$, find on which day the sales profit $W$ (yuan) is maximized and what is the maximum profit?<br/>$(3)$ If the supermarket wants the minimum value of the daily sales profit $W$ (yuan) from the 30th day to the 40th day to be $5460$ yuan, and needs to increase the selling price by $k$ yuan per lantern on that day $(0 \lt k \lt 8)$, find the value of $k$.

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