
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
At the end of the term, a certain class needs to purchase some prizes. After market research, it was found that buying $10$ pen gift boxes and $1$ water bottle costs $242$ yuan, and buying $1$ pen gift box and $10$ water bottles costs $341$ yuan. <br/>$(1)$ Can you find out the price of each pen gift box and each water bottle? (Solve using a system of linear equations) <br/>$(2)$ Based on the class situation, a total of $30$ pen gift boxes and water bottles need to be purchased. It is required that the number of pen gift boxes purchased does not exceed twice the number of water bottles purchased, and the total cost does not exceed $800. Can you calculate how many purchasing plans are possible? Which plan has the lowest cost?

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