
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 student backpack. It is known that the cost price of this backpack is $30$ yuan each. Market research shows that the daily sales quantity $y$ (in units) of this backpack is related to the selling price $x$ (in yuan) as follows: $y=-x+60$ ($30\leqslant x\leqslant 60$). Let $w$ represent the daily profit from selling this type of backpack.

(1) Find the functional relationship between $w$ and $x$.

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

(3) At what selling price should this backpack be priced to maximize the daily profit? What is the maximum profit?

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