
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
Advocating a healthy lifestyle and promoting national fitness, a certain community purchased several pieces of fitness equipment of types $A$ and $B$ last year. After investigation, it was found that the unit price of type $B$ fitness equipment is $1.5$ times that of type $A$ fitness equipment. Buying type $A$ fitness equipment for $7200$ yuan is $10$ more pieces than buying type $B$ fitness equipment for $5400$ yuan.<br/>$(1)$ What are the unit prices of type $A$ and type $B$ fitness equipment, respectively?<br/>$(2)$ If the unit prices of the two types of fitness equipment remain the same as last year, and the community plans to purchase a total of $50$ pieces of type $A$ and type $B$ fitness equipment this year, with a total cost not exceeding $21000$ yuan, how many pieces of type $A$ fitness equipment should be purchased at least?

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