
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 = An^6 + \frac{1}{2}n^5 + \frac{5}{12}n^4 + Bn^2,\\)

\\(\ldots\\)

Infer the value of \\(A+B=\\) ________.

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