
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 received a batch of clothing, with a cost of $50$ yuan per item. If they sell each item for $60$ yuan, they can sell $800$ items. If they increase the price by $5$ yuan per item, the sales volume will decrease by $100$ items.  
$(1)$ Find the sales volume and profit when the selling price is $70$ yuan.  
$(2)$ Find the function relationship between the profit $y$ (in yuan) and the selling price $x$ (in yuan), and determine at what price the maximum profit is obtained.  
$(3)$ If the store wants to make a profit of $12000$ yuan from selling this batch of clothing, what should be the selling price?

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