
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
Vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three vectors in the same plane, where $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (-2, 3)$, and $\overrightarrow{c} = (-2, m)$.

(1) If $\overrightarrow{a} \perp (\overrightarrow{b} + \overrightarrow{c})$, find $|\overrightarrow{c}|$.

(2) If $k\overrightarrow{a} + \overrightarrow{b}$ is collinear with $2\overrightarrow{a} - \overrightarrow{b}$, find the value of $k$.

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