
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 $\overrightarrow{a}=(x,-1)$, $\overrightarrow{b}=(x-2,3)$, and $\overrightarrow{c}=(1-2x,6)$.
1) If $\overrightarrow{a} \perp (2\overrightarrow{b}+ \overrightarrow{c})$, find $|\overrightarrow{b}|$.
2) If $\overrightarrow{a} \cdot \overrightarrow{b} < 0$, find the range of $x$.

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