
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
In a county-wide junior high school mathematics core literacy evaluation activity, students were given notebooks and pens as prizes. It is known that each notebook costs $3$ yuan more than each pen. The number of notebooks purchased for $390$ yuan is the same as the number of pens purchased for $300$ yuan.  
$(1)$ What are the unit prices of the notebooks and pens?  
$(2)$ If the top $50$ students in the county are each given either a notebook or a pen as a prize for this activity, in order to ensure that the total cost of purchasing the prizes does not exceed $560$ yuan, how many notebooks can be purchased at most?

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