
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 school prepared some pens and notebooks (with the same price for each pen and the same price for each notebook) as prizes for a math competition. If buying $2$ pens and $3$ notebooks costs $62$ dollars, and buying $5$ pens and $1$ notebook costs $90$ dollars.  
$(1)$ How much does it cost to buy one pen and one notebook each?  
$(2)$ If the school plans to purchase a total of $80$ prizes, including pens and notebooks, and the cost does not exceed $1100$ dollars, how many pens can the school purchase at most?

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