
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 that the arithmetic sequence $\{a_n\}$ has partial sum $S_n$ satisfying $S_2 = -1$ and $S_5 = 5$, there is another sequence $\{b_n\}$ with partial sum $T_n$ such that $b_n = (a_n + 2)\cos\left(\frac{(a_n + 2)\pi}{2}\right) + \frac{1}{a_{2n-1}a_{2n+1}}$. Determine the value of $T_{2016} + \frac{2016}{4031}$.

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