
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 $\triangle ABC$, it is known that $AB=1$, $AC=2$, and $\angle A=60^{\circ}$. If point $P$ satisfies $\overrightarrow{AP}=\overrightarrow{AB}+\lambda\overrightarrow{AC}$, and $\overrightarrow{BP}\cdot\overrightarrow{CP}=1$, then the value of the real number $\lambda$ is \_\_\_\_\_\_.

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