
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
The sums of the terms of two arithmetic progressions, each having $n$ terms, are $n^{2} + p n$ and $3 n^{2} - 2 n$. Find the condition under which the $n$-th terms of these progressions will be equal.

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