
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 &= \sin(a + b), \\ 
y &= \sin(b + c), \\
z &= \sin(c + d), \\
w &= \sin(d + a),
\end{align*}
where $a, b, c, d$ are angles in radians such that $a + b + c + d = \pi$.

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