
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 trinomials \( P_{i}(x) = x^{2} + a_{i} x + b_{i} \) for \( i = 1, \ldots, 2016 \), only one quadratic trinomial \( P_{k}(x) \) has real roots. Find all possible values of \( k \).

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