
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
Our school's sports day requires purchasing a batch of basketballs and badminton rackets as prizes. It is known that the price of a basketball is $50$ yuan more expensive than a badminton racket; purchasing $2$ basketballs and $3$ badminton rackets in total costs $250$ yuan.<br/>$(1)$ Find the price of a basketball and a badminton racket respectively.<br/>$(2)$ The school plans to purchase a total of $200$ items of these two prizes, with the quantity of basketballs being $a$ (where $a\geqslant 80$). The total cost of purchasing these two items is $W$ yuan. Please design the most cost-effective purchasing plan and calculate the minimum cost $W$.

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