
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 that \( |\vec{a}| = \sqrt{2} \), \( |\vec{b}| = 3 \), and the angle between \( \vec{a} \) and \( \vec{b} \) is \( 45^\circ \), find the range of values for \( \lambda \) such that the angle between \( \vec{a} + \lambda \vec{b} \) and \( \lambda \vec{a} + \vec{b} \) is acute.

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