
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

Suppose the function \( f(x) \) is a monotonic function on \(\mathbb{R}\) that satisfies: for any \( x, y \in \mathbb{R} \), we have \( f(x + y) = f(x) + f(y) \) and \( f(1) = 2 \). The question is: for what values of the real number \( k \) does there exist \( t > 2 \) such that
$$
f\left( k \log_{2} t \right) + f\left( \left( \log_{2} t \right)^2 - \log_{2} t - 2 \right) < 0 ?
$$

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