
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
Let  $\mathbb{R}^{+}$  be the set of positive real numbers. Find all real numbers  $a$  for which there exists a function  $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$  such that  $3(f(x))^{2}=2f(f(x))+ax^{4}$ , for all  $x \in \mathbb{R}^{+}$ .

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