
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 sequences of real numbers \(\left\{a_{i}\right\}_{i=1}^{\infty}\) and \(\left\{b_{i}\right\}_{i=1}^{\infty}\) satisfy \(a_{n+1}=\left(a_{n-1}-1\right)\left(b_{n}+1\right)\) and \(b_{n+1}=a_{n} b_{n-1}-1\) for \(n \geq 2\), with \(a_{1}=a_{2}=2015\) and \(b_{1}=b_{2}=2013\). Evaluate, with proof, the infinite sum
$$
\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right) .
$$

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