
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
The advent of the 5G era will bring about huge changes in human life. There are two models of 5G phones, $A$ and $B$, with purchase prices and selling prices as shown in the table below:

| Model | Purchase Price (¥/unit) | Selling Price (¥/unit) |
|-------|-------------------------|------------------------|
| $A$   | $3000$                  | $3400$                 |
| $B$   | $3500$                  | $4000$                 |

A certain business hall purchased a total of $A$ and $B$ phones, spending $32000$ yuan. After selling the phones, they made a total profit of $4400$ yuan.
$(1)$ How many units of $A$ and $B$ phones did the business hall purchase?
$(2)$ If the business hall purchases an additional 30 units of $A$ and $B$ phones, with the number of $B$ phones not exceeding twice the number of $A$ phones, design a plan to maximize profit when purchasing the two types of phones, and what is the maximum profit?

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