
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
Suppose $a\_1$, $a\_2$, $a\_3$, $a\_4 \in R^{+}$, the following inequalities hold: $\frac{a\_1+a\_2}{2} \geqslant \sqrt{a\_1a\_2}$, $\frac{a\_1+a\_2+a\_3}{3} \geqslant 3a\_1a\_2a\_3$, $\frac{a\_1+a\_2+a\_3+a\_4}{4} \geqslant 4a\_1a\_2a\_3a\_4$. The inferred inequality that holds is \_\_\_\_\_\_. (Note the conditions under which it holds)

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