
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
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ $(n=1,2, \cdots)$. If $\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\left\{n \in \mathbf{N}^{*} \mid \Delta\left(\Delta a_{n}\right) \geqslant-2015\right\}$.

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