
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 $|\vec{a}|=l$ and $|\vec{b}|=\sqrt{2}$, and $\vec{b}\cdot(2\vec{a}+\vec{b})=1$, the cosine of the angle between vector $\vec{a}$ and $\vec{b}$ is ______.

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