
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
$$
A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}
$$

Find the largest integer less than or equal to \(\frac{1}{A}\).

The following decimal approximation might be useful: \(0.6931 < \ln (2) < 0.6932\), where \(\ln\) denotes the natural logarithm function.

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