
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 \( n \geq 2 \) be an integer. Determine the smallest positive real number \( \gamma \) such that for any positive real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) and any real numbers \( y_{1}, y_{2}, \ldots, y_{n} \) with \( 0 \leq y_{1}, y_{2}, \ldots, y_{n} \leq \frac{1}{2} \) satisfying \( x_{1}+x_{2}+\ldots+x_{n}=y_{1}+y_{2}+\ldots+y_{n}=1 \), we have that

$$
x_{1} x_{2} \ldots x_{n} \leq \gamma\left(x_{1} y_{1}+x_{2} y_{2}+\ldots+x_{n} y_{n}\right)
$$

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