
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 store sells a type of backpack, and it is known that the cost price of this backpack is $30$ yuan each. Market research shows that the daily sales quantity $y$ (unit: pieces) of this backpack is related to the selling price $x$ (unit: yuan) as follows: $y=-x+60$ ($30\leqslant x\leqslant 60$). Let $w$ yuan be the daily profit from selling this type of backpack.

$(1)$ Find the analytical expression of the function between $w$ and $x$.

$(2)$ At what selling price should this type of backpack be set to maximize the daily profit? What is the maximum profit?

$(3)$ If the price department stipulates that the selling price of this type of backpack should not exceed $48$ yuan, and the store needs to make a daily profit of $200$ yuan from selling this type of backpack, what should be the selling price?

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