
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
Ancient Greek mathematicians from the Pythagorean school studied various polygonal numbers, such as triangular numbers 1, 3, 6, 10, ..., with the $n$-th triangular number being $\frac{n(n+1)}{2} = \frac{1}{2}n^{2} + \frac{1}{2}n$. Let the $n$-th $k$-sided polygon number be denoted as $N(n, k)$ ($k \geq 3$). Below are the expressions for the $n$-th number of some $k$-sided polygon numbers:  
Triangular numbers $N(n, 3) = \frac{1}{2}n^{2} + \frac{1}{2}n$  
Square numbers $N(n, 4) = n^{2}$  
Pentagonal numbers $N(n, 5) = \frac{3}{2}n^{2} - \frac{1}{2}n$  
Hexagonal numbers $N(n, 6) = 2n^{2} - n$  
...  
From this, we can deduce the expression for $N(n, k)$ and calculate $N(8, 12) = \_\_\_\_\_\_$.

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