
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
Find all monotonically increasing or monotonically decreasing functions  $f: \mathbb{R}_+\to\mathbb{R}_+$  which satisfy the equation  $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$  for any two numbers  $x$  and  $y$  from  $\mathbb{R}_+$ .

Hereby,  $\mathbb{R}_+$  is the set of all positive real numbers.

*Note.* A function  $f: \mathbb{R}_+\to\mathbb{R}_+$  is called *monotonically increasing* if for any two positive numbers  $x$  and  $y$  such that  $x\geq y$ , we have  $f\left(x\right)\geq f\left(y\right)$ .
A function  $f: \mathbb{R}_+\to\mathbb{R}_+$  is called *monotonically decreasing* if for any two positive numbers  $x$  and  $y$  such that  $x\geq y$ , we have  $f\left(x\right)\leq f\left(y\right)$ .

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