
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
Summer is approaching, to meet the needs of the public, an ice cream shop has launched a new boxed ice cream (divided into types $A$ and $B$, not sold separately) with a total of $50$ boxes. The cost of both types of ice cream is $15$ yuan per box. Assuming that all the purchased ice cream is sold out, and the sales volume of both types of ice cream is a positive integer. Let the selling price of type $A$ ice cream be $x$ yuan ($x \geqslant 20$, $x$ is an integer), and the total profit obtained is $y$ yuan. It is known that when the selling price $x=20$, the sales volume of type $A$ ice cream is 40 boxes. On this basis, for each increase of $1$ yuan in the selling price $x$, the sales volume decreases by 2 boxes. The selling price of type $B$ ice cream is always $30$ yuan. Let the total profit obtained from selling type $B$ ice cream be $s$ yuan.
$(1)$ Express the following quantities in algebraic expressions containing $x$:
① Profit obtained from selling type $A$ ice cream, $y=$______;
② Profit obtained from selling type $B$ ice cream, $s=______$.
$(2)$ In this sales event, the total profit obtained by the ice cream shop is $w$ yuan. Find the value of $x$ for which $w$ has the maximum value, and calculate this maximum value.
$(3)$ For each box of type $B$ ice cream sold, the ice cream shop owner donates $a$ yuan ($0 < a < 10$) to the Hope Project. If the maximum value of the total profit is $722$ yuan, find the value of $a$.

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