
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}_{i}} > 0(i=1,2,3,…,n)\\), observe the following inequalities:

\\(\dfrac{{{a}_{1}}+{{a}_{2}}}{2} \geqslant \sqrt{{{a}_{1}}{{a}_{2}}}\\);

\\(\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}}{3} \geqslant \sqrt[3]{{{a}_{1}}{{a}_{2}}{{a}_{3}}}\\);

\\(\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}}{4} \geqslant \sqrt[4]{{{a}_{1}}{{a}_{2}}{{a}_{3}}{{a}_{4}}}\\);

\\(……\\) Following this pattern, when \\(n \in \mathbb{N}^*(n \geqslant 2)\\), \\(\dfrac{{{a}_{1}}+{{a}_{2}}+\ldots +{{a}_{n}}}{n} \geqslant \)        .

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