
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
A portion of the graph of $y = g(x)$ is shown in blue below, where $g(x)$ is a quadratic function. The distance between grid lines is $1$ unit.

What is the sum of all distinct values of $x$ such that $g(g(g(x))) = 1$?

[asy]
size(150);
real g(real x) {return x^2/5 - 2*x - 4;}
draw(graph(g,-10,5), blue);
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
[/asy]

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