
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 $T$ be the set of all nonzero real numbers. The function $g : T \to T$ satisfies the following two properties:

(i) $g \left( \frac{1}{x} \right) = 2xg(x)$ for all $x \in T.$

(ii) $g \left( \frac{1}{x} \right) +  g \left( \frac{1}{y} \right) = 2 + g \left( \frac{1}{x+y} \right)$ for all $x, y \in T$ such that $x + y \in T.$

Let $n$ be the number of possible values of $g(1),$ and let $s$ be the sum of all possible values of $g(1).$ Find $n \times s.$

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