
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 fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:

| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|---------------------------------------|------------------------|
| First          | $60$                                  | $40$                                  | $1520$                 |
| Second         | $30$                                  | $50$                                  | $1360$                 |

$(1)$ Find the purchase prices of type A and type B fruits.  
$(2)$ After selling all the fruits purchased in the first two batches, the fruit store decides to reward customers by launching a promotion. In the third purchase, a total of $200$ $\text{kg}$ of type A and type B fruits are bought, and the capital invested does not exceed $3360$ $\text{元}$. Of these, $m$ $\text{kg}$ of type A fruit and $3m$ $\text{kg}$ of type B fruit are sold at the purchase price, while the remaining type A fruit is sold at $17$ $\text{元}$ per $\text{kg}$ and type B fruit is sold at $30$ $\text{元}$ per $\text{kg}$. If all $200$ $\text{kg}$ of fruits purchased in the third batch are sold, and the maximum profit obtained is not less than $800$ $\text{元}$, find the maximum value of the positive integer $m$.

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