
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 $|\vec{a}| = 3\sqrt{2}, |\vec{b}| = 4$, and the angle between $\vec{a}$ and $\vec{b}$ is $135^\circ$. Also, $\vec{m} = \vec{a} + \vec{b}, \vec{n} = \vec{a} + \lambda \vec{b}$, and $\vec{m} \perp \vec{n}$, then $\lambda =$ _______.

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