
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 a rectangular prism $ABCD-A_{1}B_{1}C_{1}D_{1}$, it is known that the distances from diagonal $AC_{1}$ to edges $DD_{1}$, $A_{1}B_{1}$, and $BC$ are $\frac{1}{m}$, $\frac{1}{n}$, and $\frac{1}{p}$ respectively. Find the length of the diagonal $AC_{1}$.

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