
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
The points \( G \) and \( I \) are on the circle with center \( H \), and \( FI \) is tangent to the circle at \( I \). The distances \( FG \) and \( HI \) are integers, and \( FI = FG + 6 \). The point \( G \) lies on the straight line through \( F \) and \( H \). How many possible values are there for \( HI \)?

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