
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 $\overrightarrow{b}$ that satisfy the conditions: $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$.

(I) Find the magnitude of $\overrightarrow{a} + \overrightarrow{b}$;

(II) If $\lambda \overrightarrow{a} - 6\overrightarrow{b}$ and $\lambda \overrightarrow{a} + \overrightarrow{b}$ are perpendicular to each other, find the value of $\lambda$.

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