
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 an equilateral triangle with side length 16. Points \(D, E, F\) are on \(CA\), \(AB\), and \(BC\), respectively, such that \(DE \perp AE\), \(DF \perp CF\), and \(BD = 14\). The perimeter of \(\triangle BEF\) can be written in the form \(a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}\), where \(a, b, c\), and \(d\) are integers. Find \(a + b + c + d\).

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