
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
On the occasion of Children's Day, a certain store purchased a batch of toys for $3000$ yuan, which sold out quickly. When they purchased the second time, the cost of each item increased by $20\%$, and the quantity purchased for $3000$ yuan was $10$ items less than the first time.  
$(1)$ Find the cost of each item for the first purchase.  
$(2)$ If the selling price of the two purchases is the same, and after selling all the toys, the profit is not less than $1700$ yuan, what should be the minimum selling price?

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