
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
The ﬁgure on the right shows the ellipse  $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$ .
Let  $R_1,R_2,R_3,$  and  $R_4$  denote those areas within the ellipse that are in the ﬁrst, second, third, and fourth quadrants, respectively. Determine the value of  $R_1-R_2+R_3-R_4$ .

[asy]
defaultpen(linewidth(0.7));
pair c=(19,98);
real dist = 30;
real a = sqrt(1998*19),b=sqrt(1998*98);
xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow);
yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow);
draw(ellipse(c,a,b));
label(" $R_1$ ",(100,200));
label(" $R_2$ ",(-80,200));
label(" $R_3$ ",(-60,-150));
label(" $R_4$ ",(70,-150));[/asy]

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