
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 of positive terms $\{a_n\}$ with partial sum $S_n$, and $2S_n = a_n^2 + a_n$ for all natural numbers n, determine the sum of the first 2016 terms of the sequence $\{C_n\}$, where $C_n = (-1)^n \frac{2a_n + 1}{2S_n}$.

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