
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 ancient Greek mathematicians of the Pythagorean school studied various polygonal numbers\\(.\\) such as triangular numbers\\(1\\) ,\\(3\\) ,\\(6\\) ,\\(10\\) , the \\(n\\)th triangular number is \\( \dfrac {n(n+1)}{2}= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n.\\) Let the \\(n\\)th \\(k\\)-gonal number be denoted by \\(N(n,k)(k\\geqslant 3)\\) . The following lists some expressions for the \\(n\\)th number in the \\(k\\)-gonal numbers:
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 the value of \\(N(20,15)\\) as \_\_\_\_\_\_.

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