
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 factory produces two types of handicrafts, $A$ and $B$. The profit from $B$ is $105$ yuan more per day than from $A$. The quantity of $A$ when the profit is $30$ yuan for $A$ and $240$ yuan for $B is equal.<br/>$(1)$ How much profit is earned from producing one piece of $A$ and one piece of $B$ respectively?<br/>$(2)$ The factory arranges for $65$ people to produce the two types of handicrafts, $A$ and $B. Each person can produce $2$ pieces of $A$ or $1$ piece of $B$ per day. Under the conditions of $(1)$, at least $5$ pieces of $B$ must be produced per day. When $5$ pieces of $B$ are produced per day, the profit per piece remains the same. If one more piece is produced, the average profit per piece decreases by $2$ yuan that day. Find the number of people producing the two types of handicrafts per day and the maximum total profit $W$ (in yuan) that can be obtained.

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