
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
Next to the ancient city of Jingzhou, a shop on "Jing Street" plans to purchase and sell two types of cultural and creative ornaments, $A$ and $B$, to tourists. It is known that the shop spends $1400$ yuan to purchase type $A$, which is twice the amount spent on type $B$ at $630$ yuan. The cost price of type $A$ is $1$ yuan more per item than the cost price of type $B$. Both types of ornaments are sold for $15$ yuan each. The plan is to purchase a total of $600$ pieces of these two types of ornaments, with the quantity of type $B$ not less than $390$ pieces and not exceeding four times the quantity of type $A$.  
$(1)$ Find the cost price per item of types $A$ and $B$.  
$(2)$ If there is only one scenario where a discount can be offered for purchasing these two types of ornaments, which is when purchasing more than $150$ pieces of type $A, the excess of type $A$ will be discounted at $40\%$ of the cost price. Let $x$ be the quantity of type $A$ ornaments purchased.  
① Find the range of values for $x$.  
② Design a plan that maximizes the profit from this purchase of ornaments and calculate the maximum profit.

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