
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
Triangle  $ABC$  is acute. Equilateral triangles  $ABC',AB'C,A'BC$  are constructed externally to  $ABC$ . Let  $BB'$  and  $CC'$  intersect at  $F$ . Let  $CC'$  intersect  $AB$  at  $C_1$  and  $AA'$  intersect  $BC$  at  $A_1$ , and let  $A_1C_1$  intersect  $AC$  at  $D$ . If  $A'F=23$ ,  $CF=13$ , and  $DF=24$ , find  $BD$ .

*2017 CCA Math Bonanza Team Round #10*

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