
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
For a given \( m \in [-\sqrt{3}, \sqrt{3}] \), find the range of the real number \( n \) such that for any \( x, y, z \in \mathbb{R} \), the inequality

\[ 
x^{2}+2y^{2}+3z^{2}+2xy+2mzx+2nyz \geqslant 0 
\]

always holds.

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