
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
Determine the maximal value of  $k$  such that the inequality  $$ \left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left(  \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right) $$  holds for all positive reals  $a, b, c$ .

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