
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
Given that \( AM^2 = CM \cdot MN \) or \( \frac{AM}{MN} = \frac{CM}{AM} \), and since \( AM = MK \), we have \( \frac{AM}{MN} = \frac{CM}{MK} \).

Consider triangles \( \triangle AMN \) and \( \triangle CMK \). With \( \angle AMN = \angle CMK \) and \( \frac{AM}{MN} = \frac{CM}{MK} \), the triangles are similar by the Angle-Angle-Angle (AAA) criterion. From this similarity, it follows that \( \angle ANC = \angle CKA \).

Now, consider triangles \( \triangle BNC \) and \( \triangle BKA \). With \( \angle B \) being common and \( \angle BNC = \angle BKA \) (since they are adjacent to equal angles), the triangles \( \triangle ABK \) and \( \triangle CBN \) are similar by the Side-Angle-Side (SAS) criterion. From this similarity, it follows that \( \frac{BK}{NB} = \frac{AB}{BC} \).

Next, consider triangles \( \triangle BNK \) and \( \triangle BCA \). Given \( \frac{BK}{NB} = \frac{AB}{BC} \) and \( \angle B \) being common, the triangles \( \triangle BNK \) and \( \triangle CBN \) are similar by the Side-Angle-Side (SAS) criterion. From this similarity, it follows that \( \angle BKN = \angle BAC \).

Thus,
\[ \angle BAC = 180^\circ - (\angle ABC + \angle BCA) = 180^\circ - (47^\circ + 64^\circ) = 69^\circ = \angle BKN \]

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