
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
Given the function $f(x)$ satisfies $f(0) = -1$, and its derivative $f'(x)$ satisfies $f'(x) > k > 1$, then among the following statements, the correct one(s) is/are \_\_\_\_\_\_.
(1) $f\left(\frac{1}{k}\right) > \frac{1}{k} - 1$; (2) $f\left(\frac{1}{k-1}\right) > \frac{1}{k-1}$; (3) $f\left(\frac{1}{k-1}\right) < \frac{2-k}{k-1}$; (4) $f\left(\frac{1}{k}\right) < f\left(\frac{1}{k-1}\right)$

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