
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  $\triangle ABC$ ,  $\angle A = 30^{\circ}$  and  $AB = AC = 16$  in. Let  $D$  lie on segment  $BC$  such that  $\frac{DB}{DC} = \frac23$  . Let  $E$  and  $F$  be the orthogonal projections of  $D$  onto  $AB$  and  $AC$ , respectively. Find  $DE + DF$  in inches.

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