
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 vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is ${120}^{{\circ }}$ and $|\overrightarrow{AB}|=|\overrightarrow{AC}|=2$, if $\overrightarrow{AP}=\lambda \overrightarrow{AB}+\overrightarrow{AC}$ and $\overrightarrow{AP}\perp \overrightarrow{BC}$, find the value of the real number $\lambda \mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}$.

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