
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 sells apples and pears. Buying $1$ kilogram of apples and $3$ kilograms of pears costs $26$ yuan, and buying $2$ kilograms of apples and $1$ kilogram of pears costs $22$ yuan. <br/>$(1)$ Find the selling price per kilogram of apples and per kilogram of pears. <br/>$(2)$ If buying a total of $15$ kilograms of apples and pears does not exceed $100$ yuan, what is the maximum number of kilograms of apples that can be purchased?

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