
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 $P$ lies on the line segment $AB$ such that $\overrightarrow{AP} = λ\overrightarrow{AB}$ where $0 \leqslant λ \leqslant 1$.

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

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