
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 fruit distributor purchased 10 boxes each of two types of fruits, $A$ and $B$, and distributed them to Store A and Store B for sale (in whole boxes). The expected profit per box of fruit at Store A and Store B is shown in the table below:

|          | $A$ Fruit / Box | $B$ Fruit / Box |
|----------|-----------------|-----------------|
| Store A  | $11$ yuan       | $17$ yuan       |
| Store B  | $9$ yuan        | $13$ yuan       |

$(1)$ If the distributor follows the plan of distributing 10 boxes to each of Store A and Store B, with 5 boxes of type $A$ fruit and 5 boxes of type $B$ fruit to each store, calculate how much profit the distributor can make.  
$(2)$ When distributing 10 boxes to each of Store A and Store B, and ensuring that Store B makes a profit of at least 115 yuan, design a distribution plan that maximizes the profit for the fruit distributor, and calculate the maximum profit in yuan.

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