
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\\), \\(…\\), with the \\(n\\)th triangular number being \\( \dfrac {n(n+1)}{2}= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n.\\) Denote the \\(n\\)th \\(k\\)-gonal number as \\(N(n,k)(k\\geqslant 3)\\). Below are expressions for some \\(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\\),
…
Deduce the expression for \\(N(n,k)\\) and calculate \\(N(10,24)=\) \_\_\_\_\_\_.

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