
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 store purchased an unspecified number of $2B$ pencils for $600$ yuan for the first time. For the second time, they purchased the same type of pencil for $600$ yuan, but this time the cost per pencil was $\frac{5}{4}$ times the cost of the first purchase, and the quantity purchased was 30 pencils less than the first time.  
$(1)$ Find the cost per pencil for the first purchase.  
$(2)$ If the profit from selling all the pencils purchased in the two transactions at the same price is not less than $420$ yuan, what is the minimum selling price per pencil?

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