
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
Since the new curriculum standard was issued in 2022, our school has attached great importance to the learning and implementation of the new curriculum standard, and has carried out "precision teaching" that integrates information technology and teaching in depth. The school plans to purchase two types of teaching equipment, A and B. It is known that the price of type A equipment is 20% higher per unit than the price of type B equipment. The number of A-type equipment purchased with 30,000 yuan is 4 more than the number of B-type equipment purchased with 15,000 yuan.  
$(1)$ Find the unit prices of A and B equipment in yuan.  
$(2)$ Our school plans to purchase a total of 50 units of the two types of equipment, requiring the number of A-type equipment to be no less than one-third of the number of B-type equipment. Assuming that $a$ units of A-type equipment are purchased and the total cost is $w$ yuan, find the functional relationship between $w$ and $a$, and design the purchasing plan with the lowest cost.

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