
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 integer  $n\geq 2$ . Find the minimum value of  $\lambda {}$ , satisfy that for any real numbers  $a_1$ ,  $a_2$ ,  $\cdots$ ,  ${a_n}$  and  ${b}$ ,  $$ \lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}. $$  

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