
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
Compute the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= 2z + 2w + zwx, \\ 
y &= 2w + 2x + wxy, \\
z &= 2x + 2y + xyz, \\
w &= 2y + 2z + yzw,
\end{align*}
and additionally assume $w = \sin^2(\theta)$ for some angle $\theta$.

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