
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
Hanmei Middle School plans to purchase Go and Chinese Chess to enrich students' extracurricular activities. If they buy $3$ sets of Go and $5$ sets of Chinese Chess, it will cost $98$ yuan; if they buy $8$ sets of Go and $3$ sets of Chinese Chess, it will cost $158$ yuan. <br/>$(1)$ Find the price of each set of Go and each set of Chinese Chess. <br/>$(2)$ Hanmei Middle School decides to purchase a total of $40$ sets of Go and Chinese Chess, with a total cost not exceeding $550$ yuan. How many sets of Go can Hanmei Middle School purchase at most?

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