
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 $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{^{\circ}}$, and $|\overrightarrow{a}|=1, |\overrightarrow{b}|=2$,
(1) Find $|\overrightarrow{2a} - \overrightarrow{b}|$;
(2) If vector $\overrightarrow{a} + \overrightarrow{b}$ and vector $\overrightarrow{a} + k\overrightarrow{b}$ are orthogonal, find the value of the real number $k$.

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