
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 of numbers \(a_{1}, a_{2}, \ldots, a_{2016}\) is a geometric progression, and the sequence \(b_{1}, b_{2}, \ldots, b_{2016}\) is an arithmetic progression. It is known that among all quadratic polynomials \(P_{i}(x)=x^{2}+a_{i} x+b_{i}\), for \(i=1, \ldots, 2016\), only one quadratic polynomial \(\mathrm{P}_{k}(x)\) has real roots. Find all possible values of \(k\).

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