
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 $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x\] for all real numbers $x$ and $y.$

Let $m$ be the number of possible values of $g(5),$ and let $t$ be the sum of all possible values of $g(5).$ Find $m \times t.$

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