
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
Given that distinct points $G$, $H$, $I$, and $J$ lie on a line, with $GH=HI=1$ and $IJ=2$, and that points $K$ and $L$ lie on a second line, parallel to the first, with $KL=2$, and another point $M$ is also on the second line such that $KM=1$, determine all possible unique values for the area of the triangle.

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