
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 purchasing 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 per item, 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/>$(1)$ When this type of children's toy is sold at the highest price per item, how many items are sold per day?<br/>$(2)$ Find the function expression of the profit $w$ (in yuan) obtained per day for this type of children's toy in terms of the selling price $x$ (in yuan).<br/>$(3)$ At what selling price does the online store maximize the profit obtained per day for 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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