
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
Rongqing Company plans to purchase desk lamps and flashlights of the same brand from a store. It is known that buying a desk lamp costs $20 more than buying a flashlight. If they spend $400 on desk lamps and $160 on flashlights, the number of desk lamps purchased is half the number of flashlights purchased.  
$(1)$ Find the cost of purchasing one desk lamp and one flashlight of this brand.  
$(2)$ After negotiations, the store offers Rongqing Company a promotion of giving a free flashlight for every desk lamp purchased. If Rongqing Company needs 2 times more flashlights than desk lamps plus 8 more flashlights, and the total cost of purchasing desk lamps and flashlights does not exceed $670, then how many desk lamps of this brand can Rongqing Company purchase at most?

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