
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 $a_i > 0$ ($i=1,2,3,\ldots,n$), observe the following inequalities: $$\frac {a_{1}+a_{2}}{2} \geq \sqrt {a_{1}a_{2}}$$; $$\frac {a_{1}+a_{2}+a_{3}}{3} \geq \sqrt[3]{a_{1}a_{2}a_{3}}$$; $$\frac {a_{1}+a_{2}+a_{3}+a_{4}}{4} \geq \sqrt[4]{a_{1}a_{2}a_{3}a_{4}}$$;  
...  
Following this pattern, when $n \in \mathbb{N}^*$ ($n \geq 2$), $$\frac {a_{1}+a_{2}+\ldots+a_{n}}{n} \geq$$\_\_\_\_\_\_ .

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