
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  $f(x)=cx(x-1)$ , 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}
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