
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 $B$ in a grid that spans 5 east-bound grids and 3 south-bound grids with certain restrictions? Here's the grid representation with omitted paths [notated by blocked points $C$ and $D$]:
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;

for(int i = 0; i <= 6; ++i)
if (i!=2 && i !=3)
draw((2*i,0)--(2*i,3));

for(int j = 0; j <= 3; ++j)
draw((0,j)--(12,j));

draw((2*2,0)--(2*2,1)); // Block half of column 2
draw((2*3,0)--(2*3,1)); // Block half of column 3

label("$A$",(0,3),NW);
label("$B$",(12,0),E);

dot("$C$",(4,2),NE);
dot("$D$",(6,2),NE);
[/asy]

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