
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 triangle with  $AB=3$  and  $AC=5$ . Select points  $D, E,$  and  $F$  on  $\overline{BC}$  in that order such that  $\overline{AD}\perp \overline{BC}$ ,  $\angle BAE=\angle CAE$ , and  $\overline{BF}=\overline{CF}$ . If  $E$  is the midpoint of segment  $\overline{DF}$ , what is  $BC^2$ ?

Let  $T = TNYWR$ , and let  $T = 10X + Y$  for an integer  $X$  and a digit  $Y$ . Suppose that  $a$  and  $b$  are real numbers satisfying  $a+\frac1b=Y$  and  $\frac{b}a=X$ . Compute  $(ab)^4+\frac1{(ab)^4}$ .


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