
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 diagrams below shows a  $2$  by  $2$  grid made up of four  $1$  by  $1$  squares. Shown are two paths along the grid from the lower left corner to the upper right corner of the grid, one with length  $4$  and one with length  $6$ . A path may not intersect itself by moving to a point where the path has already been. Find the sum of the lengths of all the paths from the lower left corner to the upper right corner of the grid.
[asy]
import graph; size(6cm); 
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,4)--(-1,2), linewidth(1.6)); 
draw((-1,4)--(1,4), linewidth(1.6)); 
draw((1,4)--(1,2), linewidth(1.6)); 
draw((-1,2)--(1,2), linewidth(1.6)); 
draw((0,4)--(0,2), linewidth(1.6)); 
draw((-1,3)--(1,3), linewidth(1.6)); 
draw((-0.5,1)--(-2.5,1), linewidth(1.6)); 
draw((-2.5,1)--(-2.5,-1), linewidth(1.6)); 
draw((-1.5,1)--(-1.5,-1), linewidth(1.6)); 
draw((-2.5,0)--(-0.5,0), linewidth(1.6)); 
draw((0.5,1)--(0.5,-1), linewidth(1.6)); 
draw((2.5,1)--(2.5,-1), linewidth(1.6)); 
draw((0.5,-1)--(2.5,-1), linewidth(1.6)); 
draw((1.5,1)--(1.5,-1), linewidth(1.6)); 
draw((0.5,0)--(2.5,0), linewidth(1.6)); 
draw((0.5,1)--(0.5,0), linewidth(4) + red); 
draw((0.5,0)--(1.5,0), linewidth(4) + red); 
draw((0.5,-1)--(1.5,-1), linewidth(4) + red); 
draw((1.5,0)--(1.5,-1), linewidth(4) + red);
draw((0.5,1)--(2.5,1), linewidth(4) + red); 
draw((-0.5,1)--(-0.5,-1), linewidth(4) + red); 
draw((-2.5,-1)--(-0.5,-1), linewidth(4) + red); [/asy]

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