
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 solutions to the following system of equations.  Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y   \displaystyle\frac{4y^2}{1+4y^2} = z   \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]

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