
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
A certain fitness equipment store has launched two discount promotions, and it is stipulated that only one of them can be chosen when shopping.  
Promotion 1: The purchased items are discounted by 20% off the original price;  
Promotion 2: The purchased items are discounted by $80 for every $300 spent at the original price. (For example: If the original price of the purchased item is $300, you can get a discount of $80, and you need to pay $220; if the original price of the purchased item is $770, you can get a discount of $160, and you need to pay $610)  
$(1)$ When buying a fitness equipment with an original price of $450, which promotion is more cost-effective? Please explain the reason;  
$(2)$ When buying a fitness equipment with an original price below $500, if the payment amount is the same when choosing Promotion 1 and Promotion 2, find the original price of such a fitness equipment;  
$(3)$ When buying a fitness equipment with an original price below $900, within what range of original prices is Promotion 2 more cost-effective than Promotion 1? Assuming the original price of such a fitness equipment is $a, please directly write down the range of values for $a$.

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