
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
Given the graph of \( y = g(x) \) shown below, where \( g(x) \) is a quadratic function, find the sum of all distinct numbers \( x \) such that \( g(g(g(x))) = -1 \).

[asy]
size(150);
import graph;
real g(real x) {return x^2/4 - x - 2;}
draw(graph(g,-7,5), red+linewidth(1));
xaxis("$x$", BottomTop(), Ticks(Label(fontsize(10)), Step=1, Size=2));
yaxis("$y$", LeftRight(), Ticks(Label(fontsize(10)), Step=1, Size=2));
label("$y=g(x)$", (4, g(4)), NE*lsf, red);
[/asy]

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