
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 values  $t$  for which there exist real numbers  $x$ ,  $y$ ,  $z$  satisfying : $3x^2 + 3xz + z^2 = 1$  ,
  $3y^2 + 3yz + z^2 = 4$ ,
  $x^2 - xy + y^2 = t$ .

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