
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
For a natural number $i \in \mathbb{N}^{*}$, let $a_{i,k} = i - 3(k-1)$ (where $k = 1, 2, 3, \ldots$), such as $a_{3,4} = 3 - 3(4-1) = -6$. For natural numbers $n, m$, when $n \geq 2, m \geq 2$, let $b(i, n) = a_{i,1} + a_{i,2} + a_{i,3} + \ldots + a_{i,n}$, $S(m, n) = b(1, n) + b(2, n) + b(3, n) + \ldots + b(m, n)$. Then, find the values of $b(10, 6)$ and $S(10, 6)$.

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