
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
If a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(\lambda x + \mu y) = \lambda f(x) + \mu f(y)$ (where $x, y, \lambda, \mu$ are real numbers), then $f(x)$ is called a linear transformation on $\mathbb{R}$. Consider the following statements:  
① $f(x) = 2x$ is a linear transformation on $\mathbb{R}$;  
② If $f(x)$ is a linear transformation on $\mathbb{R}$, then $f(kx) = kf(x)$ ($k \in \mathbb{R}$);  
③ If both $f(x)$ and $g(x)$ are linear transformations on $\mathbb{R}$, then $f(x) + g(x)$ is a linear transformation on $\mathbb{R}$;  
④ A necessary and sufficient condition for $f(x)$ to be a linear transformation on $\mathbb{R}$ is that $f(x)$ is a first-degree polynomial on $\mathbb{R}$.  
Among these, the true statements are. (Write down the numbers of all true statements)

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