
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
Xiao Mei plans to buy a bouquet of lilies and carnations to wish her mother a happy Mother's Day. It is known that buying $2$ lilies and $1$ carnation costs a total of $14$ yuan, and the price of $3$ carnations is $2$ yuan more than the price of $2$ lilies.<br/>$(1)$ Find the cost of one carnation and one lily.<br/>$(2)$ Xiao Mei plans to buy a total of $11$ lilies and carnations, with at least $2$ lilies. Let the cost of buying this bouquet of flowers be $w$ yuan, and let $x$ represent the number of carnations. Find the functional relationship between $w$ and $x$, and design a flower-buying plan to minimize the cost, stating the minimum cost.

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