
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 an equilateral triangle ABC, point P is on the line segment AB, and $\overrightarrow{AP} = \lambda \overrightarrow{AB}$ ($0 \leq \lambda \leq 1$).

(1) If the side length of the equilateral triangle is 6, and $\lambda = \frac{1}{3}$, find $|\overrightarrow{CP}|$;  
(2) If $\overrightarrow{AP} = \frac{3}{5} \overrightarrow{PB}$, find the value of $\lambda$;  
(3) If $\overrightarrow{CP} \cdot \overrightarrow{AB} \geq \overrightarrow{PA} \cdot \overrightarrow{PB}$, find the range of the real number $\lambda$.

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