
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]


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A certain dessert shop uses two types of ingredients, $A$ and $B$, to make two types of desserts, A and B, for sale. The amount of ingredients required to make each dessert is shown in the table below. The shop makes $x$ portions of dessert A and $y$ portions of dessert B, using a total of 2000 grams of ingredient $A$.

| Type of Dessert | Ingredient $A$ (grams) | Ingredient $B$ (grams) |
|-----------------|------------------------|------------------------|
| Dessert A       | 30                     | 15                     |
| Dessert B       | 10                     | 20                     |

$(1)$ Find the expression of $y$ in terms of $x$.  
$(2)$ Given that the profit per portion of dessert A is $a$ yuan ($a$ is a positive integer) and the profit per portion of dessert B is 2 yuan. Assuming that all desserts can be sold.  
① When $a=3$, if the total profit is at least 220 yuan, how many more grams of ingredient $B$ should be used at least?  
② With 3100 grams of ingredient $B$ available, in order to make a profit of 450 yuan and minimize waste of raw materials, how much should the profit per portion of dessert A be set at?

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