
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
Huidong Shuangyue Bay Beach is an ideal camping site, where many tourists from all over the world choose to camp. An outdoor equipment store is preparing to purchase a batch of tents for sale. It is known that the purchase price of each regular tent is $150, and the purchase price of each sunshade tent is twice that of a regular tent. The outdoor equipment store allocates $9000 for the purchase of sunshade tents and regular tents. Let $x$ represent the number of sunshade tents purchased and $y$ represent the number of regular tents purchased.  
$(1)$ Find the analytical expression of the relationship between $y$ and $x$.  
$(2)$ The selling price of a sunshade tent is set at $380 per tent, and the selling price of a regular tent is set at $180 per tent. If the number of regular tents purchased is not less than 12 and does not exceed the number of sunshade tents purchased, devise a purchasing plan to maximize the profit obtained when selling all the purchased tents, and calculate the maximum profit.

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