
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
Points $E,$ $F,$ $G,$ and $H$ are equally spaced along a line such that $EF = FG = GH.$  A point $Q$ is located so that $\cos \angle EQG = \frac{3}{5}$ and $\cos \angle FQH = \frac{1}{5}.$  Determine $\sin (2 \angle FQG).$

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