
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 positive numbers $x_{i}(i=1,2,3)$ satisfy $x_{1}+x_{2}+x_{3}=4$. For any $i$ and $j$ in $\{1,2,3\}$, $i\neq j$, it is always true that $2{x}_{i}^{2}+2{x}_{j}^{2}-5{x}_{i}{x}_{j}≤0$. Then find the minimum value of the product $x_{1}x_{2}x_{3}$.

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