
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $30^{\circ}$ between them, and $|\overrightarrow{a}|=\sqrt{3}$, $|\overrightarrow{b}|=1$,

$(1)$ Find the value of $|\overrightarrow{a}-2\overrightarrow{b}|$

$(2)$ Let vector $\overrightarrow{p}=\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{q}=\overrightarrow{a}-2\overrightarrow{b}$, find the projection of vector $\overrightarrow{p}$ in the direction of $\overrightarrow{q}$

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