
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
Let \(ABCD\) be a rectangle with \(AB = 6\) and \(BC = 4\). Let \(E\) be the point on \(BC\) with \(BE = 3\), and let \(F\) be the point on segment \(AE\) such that \(F\) lies halfway between the segments \(AB\) and \(CD\). If \(G\) is the point of intersection of \(DF\) and \(BC\), find \(BG\).

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