
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 company is preparing to purchase two types of materials, $A$ and $B$, to make two types of crafts, $X$ and $Y$. It is known that for every one unit of material $A$ purchased, it costs $40$ yuan less than one unit of material $B$. Additionally, purchasing $2$ units of material $A$ and $3$ units of material $B$ costs a total of $420$ yuan. <br/>$(1)$ What is the cost of one unit of material $A$ and one unit of material $B$ respectively?<br/>$(2)$ If the purchased materials can be used to make a total of $560$ crafts of types $X$ and $Y$, and the amount of material $A$ and $ material $B$ required to make one unit of craft $X$ and one unit of craft $Y$ is shown in the table below.<br/>

| Type of Craft | Material $A$ per unit | Material $B$ per unit |
|---------------|-----------------------|-----------------------|
| $X$           | $1$                   | $2$                   |
| $Y$           | $3$                   | $2$                   |

If the selling price of craft $X$ is $360$ yuan per unit and the selling price of craft $Y$ is $450$ yuan per unit. According to market demand, the quantity of craft $X$ should not exceed $180$ units. How should the production plan be arranged to maximize the profit obtained? Also, calculate the maximum total profit.

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