
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 an integer  $k > 1$  be given. For each integer  $n > 1$ , we put  

\[f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)\] 

where  $p_1, p_2, \ldots, p_r$  are all distinct prime divisors of  $n$ . Find all values  $k$  for which the sequence  $\{x_m\}$  defined by  $x_0 = a$  and  $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$  is bounded for all integers  $a > 1$ .

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