
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 $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (x, 1)$,
1. If $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$ forms an acute angle, find the range of $x$.
2. Find the value of $x$ when $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$.

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