
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 $\triangle ABC$ be a right triangle with $\angle ABC = \dfrac{\pi}{2}$. The hypotenuse $AC$ has length $20$. Points $D$ and $E$ are on $AC$ such that $AD = 12$ and $DC = 8$. A perpendicular line is drawn from $B$ to $AC$, intersecting at point $F$. Let the distance $DF = x$ and the length of $BF$ be $2 \sqrt{21}$. Find the value of $x$.

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