
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
Let  $n\geq 2$  an integer. Find the least value of  $\gamma$  such that for any positive real numbers  $x_1,x_2,...,x_n$  with  $x_1+x_2+...+x_n=1$  and any real  $y_1+y_2+...+y_n=1$  and  $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$  the following inequality holds:  $$ x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right) $$  

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