
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
The mathematicians of the ancient Greek Pythagorean school studied various polygonal numbers, such as triangular numbers $(1)$, $(3)$, $(6)$, $(10)$, $(…)$, with the $n$-th triangular number being $\dfrac {n(n+1)}{2}= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n$. Let $N(n,k)$ represent the $n$-th $k$-sided polygonal number $(k\geqslant 3)$. The following lists some expressions for the $n$-th $k$-sided polygonal number:
Triangular numbers $N(n,3)= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n$
Square numbers $N(n,4)=n^{2}$
Pentagonal numbers $N(n,5)= \dfrac {3}{2}n^{2}- \dfrac {1}{2}n$
Hexagonal numbers $N(n,6)=2n^{2}-n$
$…$
We can conjecture the expression for $N(n,k)$ and use it to calculate $N(8,12)=$ _____ .

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