
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 $\left| \vec{a} \right|=2, \left| \vec{b} \right|=2$ and the angle between $\vec{a}$ and $\vec{b}$ is $45^\circ$, and $\lambda \vec{b}-\vec{a}$ is perpendicular to $\vec{a}$, then the real number $\lambda =$ \_\_\_\_\_\_\_\_\_\_.

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