
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 that \( f(x) \) is an even function and \( f(x) = x - \lfloor x \rfloor \) for \( x \leq 0 \) (where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \)), if the equation \( f(x) = kx + k \) (where \( k > 0 \)) has exactly three distinct real roots, find the range of the real number \( k \).

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