
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 all functions  $f:\mathbf{R} \rightarrow \mathbf{R}^+$  such that for all real numbers  $x,y$  the following conditions hold: $\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) 
ii. & f(-x) = f(x-1)
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$ 

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