
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
Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{3} + \sqrt{5} + \sqrt{11}}$, and write your answer in the form 
\[
\frac{A\sqrt{3} + B\sqrt{5} + C\sqrt{11} + D\sqrt{E}}{F},
\]
where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$?

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