
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  $x_0, x_1, x_2 \dots$  be a sequence of positive real numbers such that for all  $n \geq 0$ ,   $$ x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2} $$   For which values of  $x_0$  is this sequence bounded?

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