
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
Given $f(x) = x^2 + ax$, where $\{x | f(x) = 0, x \in \mathbb{R}\} = \{x | f(f(x)) = 0, x \in \mathbb{R}\} \neq \emptyset$, determine the range of all real numbers $a$ that satisfy the condition.

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