
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
Let  $A$  be the locus of points  $(\alpha, \beta, \gamma)$  in the  $\alpha\beta\gamma$ -coordinate space that satisfy the following properties:**(I)**  We have  $\alpha$ ,  $\beta$ ,  $\gamma > 0$ .**(II)**  We have  $\alpha + \beta + \gamma = \pi$ .**(III)**  The intersection of the three cylinders in the  $xyz$ -coordinate space given by the equations
\begin{eqnarray*}
y^2 + z^2 & = & \sin^2 \alpha 
z^2 + x^2 & = & \sin^2 \beta 
x^2 + y^2 & = & \sin^2 \gamma
\end{eqnarray*}
is nonempty.  

Determine the area of  $A$ .

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