
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
I have two cents and Bill has \( n \) cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e., 10 cents for 12 pencils). Bill notes that he can spend all \( n \) of his cents on some combination of pencil packages to get \( P \) pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all \( n+2 \) of his cents on some combination of pencil packages to get fewer than \( P \) pencils. What is the smallest value of \( n \) for which this is possible?

Note: Both times Bill must spend all of his cents on pencil packages, i.e., have zero cents after either purchase.

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