
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
Consider the set \( T \) of all nonzero real numbers excluding \(-1\) and \(1\). Define a function \( g: T \to T \) that satisfies:

(i) \( g(x + y) = g(x) + g(y) \) for all \( x, y \in T \) such that \( x + y \in T \).

(ii) \( g\left(\frac{1}{\sin x}\right) + g\left(\frac{1}{\sin y}\right) = \sin x \sin y \cdot g\left(\frac{1}{\sin(x+y)}\right) \) for all \( x, y \in T \) such that \( \sin x + \sin y \neq 0 \).

Let \( p \) be the number of possible values of \( g(\frac{1}{\sin 1}) \), and let \( q \) be the sum of all possible values of \( g(\frac{1}{\sin 1}) \). Find \( p \times q \).

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