
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
Distinct points A, B, C, D, and G lie on a line, with AB = BC = CD = 1 and DG = 2. Points E and F lie on another line, parallel to the first, with EF = 2.5. Additionally, point H is located such that line HE is perpendicular to both lines containing A, B, C, D, E, F, and G, and HE = 1. Determine the number of possible values for the area of a triangle formed by three of the eight points.

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