
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 the vectors $\mathbf{a} = (1,2)$ and $\mathbf{b} = (2, -2)$:
1. If $\mathbf{c} = 4\mathbf{a} + \mathbf{b}$, find $(\mathbf{b} \cdot \mathbf{c})\mathbf{a}$;
2. If $\mathbf{a} + \lambda \mathbf{b}$ is perpendicular to $\mathbf{a}$, find the value of $\lambda$.

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