
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
For each  $r\in\mathbb{R}$  let  $T_r$  be the transformation of the plane that takes the point  $(x, y)$  into the point  $(2^r x; r2^r x+2^r y)$ . Let  $F$  be the family of all such transformations (i.e.  $F = \{T_r : r\in\mathbb{R}\}$ ). Find all curves  $y = f(x)$  whose graphs remain unchanged by every transformation in  $F$ .

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