
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
For an integer  $n\ge 2$  find all  $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$  so that 
(a)  $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b)  $\sum_{k=1}^n (a_k+b_k)=2n;$ (c)  $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$ 

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