
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 diagram, cube \(A B C D E F G H\) has edges of length \(a\). Points \(M, N\) and \(P\) are on edges \(G C\), \(F B\), and \(E A\), respectively, so that \(M C=\frac{1}{2} a\), \(N B=\frac{1}{3} a\), and \(P A=\frac{1}{4} a\). If the plane defined by points \(M, N\), and \(P\) intersects \(H D\) at point \(T\), determine the length \(D T\).

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