
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
In the cube \(ABC D-A_{1} B_{1} C_{1} D_{1}\), points \(E\), \(F\), and \(G\) are located on edges \(A A_{1}\), \(A_{1} D_{1}\), and \(D_{1} C_{1}\), respectively, with \(E\) being the midpoint of \(A A_{1}\). Given that 
$$
\frac{D_{1} F}{D_{1} A_{1}} = \frac{D_{1} G}{D_{1} C_{1}} = \frac{1}{3},
$$
determine the tangent of the angle between the line of intersection \(m\) of the plane \(E F G\) with the plane \(A_{1} B_{1} C D\) and the plane \(A B C D\).

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