
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
How many 12-step paths are there from $E$ to $G$ which pass through $F$ and then $H$?
[asy]
size(4cm, 4cm);
int w = 7; int h = 6;
int i;
pen p = fontsize(9);
for (i=0; i<h; ++i) { draw((0,i) -- (w-1,i)); }
for (i=0; i<w; ++i) { draw((i, 0)--(i,h-1)); }
label("G", (w-1,0), SE, p);
label("E", (0,h-1), NW, p);
label("F", (4,4), NE, p);
label("H", (5,2), NE, p);
[/asy]

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