
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
From July 28th to August 8th, 2023, the 31st World University Games will be held in Chengdu. "Be a good host and warmly welcome guests." A well-known snack shop in Chengdu plans to purchase two types of ingredients, $A$ and $B$, to make snacks. It is known that purchasing 1 kilogram of ingredient $A$ and 1 kilogram of ingredient $B$ costs a total of 68 yuan, and purchasing 5 kilograms of ingredient $A$ and 3 kilograms of ingredient $B$ costs a total of 280 yuan. <br/>$(1)$ Find the unit prices of ingredients $A$ and $B$; <br/>$(2)$ The snack shop plans to purchase a total of 36 kilograms of the two ingredients, with the amount of ingredient $A$ purchased being at least twice the amount of ingredient $B$ purchased. When purchasing how many kilograms of ingredients $A$ and $B$ respectively, the total cost is minimized? And what is the minimum total cost?

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