
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 \(\{a_{n}\}\) satisfies \(\frac{1}{a_{n} a_{n+1}}+\frac{1}{a_{n} a_{n+2}}+\frac{1}{a_{n+1} a_{n+2}}=1, \quad (n \in \mathbf{Z}_{+})\) and \(a_{1}+a_{3}=6\), the terms \(a_{1}, a_{2}, a_{3}\) are monotonically increasing and form a geometric sequence, find the value of \([S_{2014}]\), where \(S_{n}\) is the sum of the first \(n\) terms of \(\{a_{n}\}\).

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