
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 $|\overrightarrow{a}|=1$, $\overrightarrow{a}•\overrightarrow{b}=\frac{1}{4}$, and $(\overrightarrow{a}+\overrightarrow{b})\cdot (\overrightarrow{a}-\overrightarrow{b})=\frac{1}{2}$. Find:<br/>
$(1)$ The value of $|\overrightarrow{b}|$.<br/>
$(2)$ The cosine value of the angle between vectors $\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$.

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