
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 $\left[ x \right]$ represents the largest integer not greater than $x$, for example $\left[ 3 \right]=3$, $\left[ 1.2 \right]=1$, $\left[ -1.3 \right]=-2$. A sequence $\left\{ {{a}_{n}} \right\}$ satisfies ${{a}_{1}}=1$, ${{a}_{n+1}}=a_{n}^{2}+{{a}_{n}}$, find the value of $\left[ \frac{{a}_{1}}{{a}_{1}+1}+ \frac{{a}_{2}}{{a}_{2}+1}+...+ \frac{{a}_{2016}}{{a}_{2016}+1} \right]$.

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