
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 $a=(1,2)$, $b=(2,-2)$.

$(1)$ Let $c=4a+b$, find $(b\cdot c)a$;

$(2)$ If $a+\lambda b$ is perpendicular to $a$, find the value of $\lambda$;

$(3)$ If $a+\lambda b$ is parallel to $a$, find the value of $\lambda$;

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