
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
Moving only south and east along the line segments, how many paths are there from $A$ to $B$? Consider a grid that is extended to 4 rows and 10 columns. Two columns are fully blocked at the third row from the top, and one column is partially blocked at the second row from the top.

[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i!=5 && i !=6 && i!= 7)
draw((2i,0)--(2i,4));
for(int j = 0; j <= 4; ++j)
draw((0,j)--(20,j));

draw((2*5,0)--(2*5,2));
draw((2*6,0)--(2*6,2));
draw((2*7,0)--(2*7,3));

label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]

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