
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
During the Dragon Boat Festival, Teacher Li bought two types of zongzi gift boxes at the supermarket. It is known that buying 3 boxes of red date zongzi costs the same as buying 4 boxes of egg yolk zongzi; buying 1 box of red date zongzi and 2 boxes of egg yolk zongzi together costs $100.  
$(1)$ Find the unit prices of these two types of zongzi gift boxes;  
$(2)$ Li bought a total of 8 boxes of zongzi gift boxes for less than $300. In one box of red date zongzi, there are 10 zongzi, and in one box of egg yolk zongzi, there are 6 zongzi. If he distributes the zongzi to 55 students and 10 subject teachers, each person can get at least one zongzi. Based on the above information, find Li's purchasing plan and the total amount spent.

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