
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 numbers \( t \) such that there exist non-negative real numbers \( x, y, z \) satisfying
$$
\left\{\begin{array}{l}
3x^{2}+3zx+z^{2}=1, \\
3y^{2}+3yz+z^{2}=4, \\
x^{2}-xy+y^{2}=t .
\end{array}\right.
$$

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