
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]


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A certain gift shop is preparing to purchase two types of souvenirs, $A$ and $B$. The cost of purchasing 9 souvenirs of type $A$ is the same as the cost of purchasing 7 souvenirs of type $B$. Each souvenir of type $A$ costs $20$ yuan less than each souvenir of type $B$. Please answer the following questions:<br/>$(1)$ What is the cost of purchasing each type of souvenir?<br/>$(2)$ If the gift shop purchases 5 more souvenirs of type $B$ than twice the number of souvenirs of type $A$, and the number of souvenirs of type $A$ is at least 18, and the total cost of purchasing both types of souvenirs does not exceed 5450 yuan, what are the possible purchasing plans for the gift shop?

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