
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
Let $\[x\]$ denote the largest integer not greater than $x$, for example, $\[3\] = 3$, $\[1.2\] = 1$, $\[-1.3\] = -2$. Given the sequence ${a\_n}$ satisfies $a\_1 = 1$, $a\_{n+1} = a\_n^2 + a\_n$, find the value of:

$$
\[ \frac {a\_1}{a\_1+1} + \frac {a\_2}{a\_2+1} + ... + \frac {a\_{2016}}{a\_{2016}+1} \]
$$

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