
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 equilateral triangle $\triangle ABC$, point $P$ lies on segment $AB$ such that $\overrightarrow{AP} = \lambda \overrightarrow{AB}$. If $\overrightarrow{CP} \cdot \overrightarrow{AB} = \overrightarrow{PA} \cdot \overrightarrow{PB}$, find the value of the real number $\lambda$.

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