
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 $S_{k}=1^{k}+2^{k}+3^{k}+…+n^{k}$, when $k=1$, $2$, $3…$, observe the following equations:
$S_{1}= \frac {1}{2}n^{2}+ \frac {1}{2}n$, $S_{2}= \frac {1}{3}n^{3}+ \frac {1}{2}n^{2}+ \frac {1}{6}n$, $S_{3}= \frac {1}{4}n^{4}+ \frac {1}{2}n^{3}+ \frac {1}{4}n^{2}$,
$S\;\_{4}= \frac {1}{5}n^{5}+ \frac {1}{2}n^{4}+ \frac {1}{3}n^{3}- \frac {1}{30}n$, $S\_{5}= \frac {1}{6}n^{6}+An^{5}+Bn^{4}- \frac {1}{12}n^{2}$, $…$,
It can be inferred that $A-B=$ \_\_\_\_\_\_.

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