
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 $|\vec{a}|= \sqrt{3}, |\vec{b}|=2$, and the angle between vector $\vec{a}$ and vector $\vec{b}$ is $150^{\circ}$.

$(1)$ Find: $| \vec{a} -2 \vec{b} |$;

$(2)$ If $( \vec{a} +3\lambda \vec{b} )\perp( \vec{a} +\lambda \vec{b} )$, find the value of the real number $\lambda$.

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