
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
Let $S_k = 1^k + 2^k + 3^k + \ldots + n^k$, for $k=1, 2, 3, \ldots$, observe the following equations:  
$S_1 = \frac{1}{2}n^2 + \frac{1}{2}n$, $S_2 = \frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n$, $S_3 = \frac{1}{4}n^4 + \frac{1}{2}n^3 + \frac{1}{4}n^2$,  
$S_4 = \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3 - \frac{1}{30}n$, $S_5 = \frac{1}{6}n^6 + An^5 + Bn^4 - \frac{1}{12}n^2$, $\ldots$,  
It can be inferred that $A-B = \boxed{\frac{1}{12}}$.

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