
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 the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is $90^{\circ}$, $|\overrightarrow{AB}|=2$, $|\overrightarrow{AC}|=1$, and $\overrightarrow{AM}=\lambda \overrightarrow{AB}+\mu \overrightarrow{AC} (\lambda, \mu \in \mathbb{R})$, and $\overrightarrow{AM} \cdot \overrightarrow{BC}=0$, then the value of $\frac{\lambda}{\mu}$ is ______.

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