
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 \( \left\{ \frac{1}{(n+1) \sqrt{n} + n \sqrt{n+1}} \right\} \) be a sequence and \( S_n \) be the sum of the first \( n \) terms of this sequence. How many rational numbers are there in the first 2016 terms of the sequence \( \left\{ S_n \right\} \)?

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