
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
A semicircle with diameter $AB$ equal to 2 has a perpendicular line through $A$ in the plane of the circle. A point $S$ is chosen on this perpendicular line such that $AS = AB$. Let $C$ be a moving point on the semicircle, and let $M$ and $N$ be the projections of $A$ onto $SB$ and $SC$, respectively. When the volume of the triangular pyramid $S-AMN$ is at its maximum, what is the sine of the angle between $SC$ and the plane $ABC$?

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