
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 $D$ is the midpoint of side $BC$ of triangle $ABC$, and point $P$ satisfies $\overrightarrow{PA} + \overrightarrow{BP} + \overrightarrow{CP} = 0, \overrightarrow{AP} = \lambda \overrightarrow{PD}$, then the value of the real number $\lambda$ is \_\_\_\_\_\_.

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