
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
Find all the real number sequences  $\{b_n\}_{n \geq 1}$  and  $\{c_n\}_{n \geq 1}$  that satisfy the following conditions:
(i) For any positive integer  $n$ ,  $b_n \leq c_n$ ;
(ii) For any positive integer  $n$ ,  $b_{n+1}$  and  $c_{n+1}$  is the two roots of the equation  $x^2+b_nx+c_n=0$ .

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