
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
For \( t \in [0, \pi] \), let \( S = \sin^{2} x + \sin^{2} y + \left(8 \sin^{2} t + 2\right) \sin x \sin y \). The maximum value of \( S \) is denoted as \( M(t) \) and the minimum value is denoted as \( N(t) \), where \( x \) and \( y \) are non-negative real numbers and \( x + y = t \). Let \( g(t) = M(t) - N(t) \). Find the maximum value of \( g(t) \) and the corresponding value of \( t \).

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