
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
The sequence  $ (x_n)$ ,  $ n\in\mathbb{N}^*$  is defined by  $ |x_1|<1$ , and for all  $ n \ge 1$ , \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for  $ x_1$  so that each  $ x_n > 0$ .
(b) Is this sequence periodic? And why?

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