
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
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden. 
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]

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