
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 $\overset{→}{a}$ and $\overset{→}{b}$ is $30^{\circ}$, and $|\overset{→}{a}| = \sqrt{3}$, $|\overset{→}{b}| = 1$,

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

(2) Let vector $\overset{→}{p} = \overset{→}{a} + 2\overset{→}{b}$, $\overset{→}{q} = \overset{→}{a} - 2\overset{→}{b}$, find the projection of vector $\overset{→}{p}$ onto $\overset{→}{q}$

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