
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 two vectors $ \vec{a}=(1,2) $ and $ \vec{b}=(x,1) $:  
(1) If the angle between $ \vec{a} $ and $ \vec{b} $ is acute, find the range of $x$.  
(2) When $\vec{a}+2\vec{b}$ is orthogonal to $2\vec{a}-\vec{b}$, find the value of $x$.

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