
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
Suppose  $f : \mathbb{R} \longrightarrow \mathbb{R}$  be a function such that
\[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\]
for all real  $x$ . Find  $f (2)$  and all possible values of  $f (1)$ . For each value of  $f (1)$ , construct a function achieving it and satisfying the given equation.

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