
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 \( f(x) = c x (x - 1) \), where \( c \) is a positive real number. We use \( f^{n}(x) \) to denote the polynomial obtained by composing \( f \) with itself \( n \) times. For every positive integer \( n \), all the roots of \( f^{n}(x) \) are real. What is the smallest possible value of \( c \)?

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