
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 $$|\overrightarrow{a}|=1$$ and $$|\overrightarrow{b}|=2$$.

(1) If $$\overrightarrow{a} \parallel \overrightarrow{b}$$, find $$\overrightarrow{a} \cdot \overrightarrow{b}$$;  
(2) If $$\overrightarrow{a} - \overrightarrow{b}$$ is perpendicular to $$\overrightarrow{a}$$, find the value of $k$ for which $$(k\overrightarrow{a} - \overrightarrow{b}) \perp (\overrightarrow{a} + 2\overrightarrow{b})$$.

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