
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
According to the materials, complete the tasks.

如何设计雪花模型材料采购方案? (How to design a material procurement plan for snowflake models?)

- Material 1: The school organized students to participate in the production of two types of snowflake models, A and B. Each snowflake model requires two types of pipe materials, long and short. A student used 6 long pipes and 48 short pipes to make 1 model of type A and 1 model of type B. It is known that the ratio of long to short pipes required to make one model of type A and B is 1:7 and 1:9, respectively.

- Material 2: The school spent $320 in May to purchase long pipes from a store, which was $80 less than the number of short pipes purchased for $200. The selling price of short pipes is $a per pipe, and the selling price of long pipes is $2a per pipe. Starting from June 1st, for every 3 long pipes purchased, 1 short pipe is given for free. The store has limited inventory, with only 267 long pipes and 2130 short pipes left.

- Material 3: In June, the school has a budget of $1280 and wants to purchase a certain number of long and short pipes from the store to make all models of type A and B (no materials left), using the entire budget.

**Problem Solving**

- Task 1: Analyze the structure of the snowflake model. Determine the number of long and short pipes needed to make one model of type A and B.

- Task 2: Determine the procurement cost. Find the value of $a$ and calculate the material procurement cost if only one model of type A is made in June.

- Task 3: Develop a procurement plan. Find all procurement plans that meet the conditions and identify which plan yields the highest total number of snowflakes.

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