
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 $H$? Assume the points are placed as follows on a grid: $E$ at the top-left corner, $F$ 3 steps right and 2 steps down from $E$, $H$ 2 steps right and 2 steps down from $F$, and $G$ 3 steps right from $H$ at the bottom-right corner.

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