
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 \( f(x) = \left\{ \begin{array}{ll} x - [x], & x \leq 0, \\ f(x - 1), & x > 0, \end{array} \right. \)
where \([x]\) represents the greatest integer less than or equal to \( x \). If the equation \( f(x) = kx + k \) (with \( k > 0 \)) has three distinct real roots, determine the range of the real number \( k \).

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