
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
Given two skew lines $a$ and $b$ that form an angle $\theta$, with their common perpendicular line $A^{\prime} A$ having a length of $d$. Points $E$ and $F$ are taken on lines $a$ and $b$ respectively. Let $A^{\prime} E = m$ and $A F = n$. Find the distance $E F$. ($A^{\prime}$ lies on line $a$ perpendicular to $A$, and $A$ lies on line $b$ perpendicular to $A^{\prime}$).

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