
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 vector $ \overrightarrow{AB} $ and $ \overrightarrow{AC} $ is $60^{\circ}$, and $| \overrightarrow{AB} |=2$, $| \overrightarrow{AC} |=4$, if $ \overrightarrow{AP}= \overrightarrow{AB}+\lambda \overrightarrow{AC} $, and $ \overrightarrow{AP} \perp \overrightarrow{BC} $, then the value of the real number $ \lambda $ is __________.

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