
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 graph of \( y = g(x) \), with \( 1 \) unit between grid lines, where \( g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \), defined only on the shown domain. 

Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.

[asy]
size(150);
real f(real x) {return (x-4)*(x-2)*x*(x+2)*(x+4)*(x+6)/720-2.5;}
draw(graph(f,-6.5,6.5), red+linewidth(1));
xaxis("$x$",EndArrow);
yaxis("$y$",EndArrow);
real[] ys = {-3, -2.5, -2};
for(real y : ys) draw((-6.5,y)--(6.5,y), dashed+green);
[/asy]

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