
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
A certain middle school is conducting an interdisciplinary comprehensive practice activity and needs to prepare two types of straws, $A$ and $B$, for which the school plans to purchase from a supermarket. After conducting a survey, the relevant data obtained is summarized in the following table:

| Purchase Quantity (unit: packs) | Total Cost (unit: yuan) |
|---------------------------------|-------------------------|
| Type $A$ straws                 | Type $B$ straws         | 
| $12$                            | $15$                    | $171$                   |
| $24$                            | $28$                    | $332$                   |

$(1)$ What is the cost per pack for type $A$ and type $B$ straws, respectively?  
$(2)$ The school decides to purchase a total of $100$ packs of type $A$ and type $B$ straws, with a total cost not exceeding $600$ yuan. How many packs of type $A$ straws can the school purchase at most?

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