
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 a real number  $ c > 0$ , a sequence  $ (x_n)$  of real numbers is defined by  $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$  for  $ n \ge 0$ . Find all values of  $ c$  such that for each initial value  $ x_0$  in  $ (0, c)$ , the sequence  $ (x_n)$  is defined for all  $ n$  and has a finite limit  $ \lim x_n$  when  $ n\to \plus{} \infty$ .

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