
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
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio  $2$  to  $1$ .  How many integer values of  $k$  are there such that  $0<k\leq 2007$  and the area between the parabola  $y=k-x^2$  and the  $x$ -axis is an integer?

[asy]
import graph;
size(300);
defaultpen(linewidth(0.8)+fontsize(10));
real k=1.5;
real endp=sqrt(k);
real f(real x) {
return k-x^2;
}
path parabola=graph(f,-endp,endp)--cycle;
filldraw(parabola, lightgray);
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));
label("Region I", (0,2*k/5));
label("Box II", (51/64*endp,13/16*k));
label("area(I) =  $\frac23$ \,area(II)",(5/3*endp,k/2));
[/asy]

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