
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 the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, $\overrightarrow{c}=λ\overrightarrow{a}-2\overrightarrow{b}$.  
$(1)$ Find $|{3\overrightarrow{a}-\overrightarrow{b}}|$;  
$(2)$ When $\overrightarrow{b}⊥\overrightarrow{c}$, find the value of $\lambda$.

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