
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 sells a type of children's toy. The cost price is $30$ yuan per item, and the pricing department stipulates that the profit margin for each children's toy sold should not exceed $50\%$ of the cost price. During the sales process, it was found that when the selling price is $35$ yuan, 350 items can be sold per day. If the selling price is increased by $5$ yuan, the daily sales volume decreases by 50 items. Let the selling price be $x$ yuan (the selling price is not less than $35$ yuan). <br/>$(1)$ Find the function expression between the profit $w$ (yuan) obtained per day and the selling price $x$ (yuan); <br/>$(2)$ At what price does the online store make the maximum profit per day selling this type of children's toy, and what is the maximum profit?

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