
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  $(x_n)_{n\ge 0}$  a sequence of real numbers defined by  $x_0>0$  and  $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$ . Compute  $\lim_{n\to \infty}x_n$  and  $\lim_{n\to \infty} \frac{x_n^3}{n^2}$ .

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