
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 stationery store is preparing to sell two types of water pens, A and B, with the cost and profit per unit as shown in the table below:

|            | Water Pen A | Water Pen B |
|------------|-------------|-------------|
| Cost (per unit in yuan) | $a$ | $a+5$ |
| Profit (per unit in yuan) | $2$ | $3$ |

It is known that the amount spent on purchasing water pen A is 400 yuan, and the amount spent on purchasing water pen B is 800 yuan. 

$(1)$ Find the cost per unit of water pen A and water pen B.

$(2)$ If the stationery store plans to purchase a total of 300 water pens, considering customer demand, and requiring the quantity of water pen A purchased to be at least 4 times the quantity of water pen B purchased, determine how the store can maximize profit from the purchase and what the maximum profit would be.

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