
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 sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{3}$, and for any $n \in \mathbf{N}^{*}, a_{n+1}=a_{n}^{2}+a_{n}$. Determine the integer part of $\sum_{n=1}^{2016} \frac{1}{a_{n}+1}$.

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