
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 $ABC$, given: $\angle ACB = 60^\circ$, $\angle ABC = 45^\circ$. Point $K$ is taken on the extension of $AC$ beyond vertex $C$ such that $AC = CK$. Point $M$ is taken on the extension of $BC$ beyond vertex $C$ such that triangle with vertices $C, M,$ and $K$ is similar to the original triangle. Find $BC: MK$ if it is known that $CM: MK < 1$.

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