
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 Pythagorean school in ancient Greece studied various polygonal numbers, such as triangular numbers $1$, $3$, $6$, $10$, $…$, where the $n$-th triangular number is $\frac{n(n+1)}{2}=\frac{1}{2}n^{2}+\frac{1}{2}n$. Denote the $n$-th $k$-gonal number as $N(n,k)\ (k\geqslant 3)$. Some expressions for the $n$-th $k$-gonal number are listed below:
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 $N(n,6)=2n^{2}-n$,
$…$
One can conjecture the expression for $N(n,k)$ and use it to calculate $N(20,32)=\_\_\_\_\_\_.$

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