
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
Find all real  $x\geq-1$  such that for all  $a_1,...,a_n\geq1$ , where  $n\geq2$   the following inequality holds   $$ \frac{a_1+x}{2}*\frac{a_2+x}{2}*...*\frac{a_n+x}{2}\leq\frac{a_1a_2...a_n+x}{2} $$  

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