
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, with $1$ unit between grid lines and $g(x)$ defined only on the domain shown. Find the most positive integer value $c$ for which the equation $g(x) = c$ has exactly 4 solutions.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {
label("Re",(xright,0),SE);
label("Im",(0,ytop),NW);
} else {
label("x",(xright+0.4,-0.5));
label("y",(-0.5,ytop+0.2));
}

ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) { TicksArrx.push(i); }
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) { TicksArry.push(i); }
}

if(usegrid) {
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);
}

if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-6,6,-7,7);
real g(real x) {return (x-4)*(x-2)*(x)*(x+2)*(x+4)/255 - 5;}
draw(graph(g,-5.5,5.5,operator ..), blue);
[/asy]

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