
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 $\mathbf{a}=(-2,2)$ and $\mathbf{b}=(5,k)$.

(1) Find the value of the real number $k$ if $\mathbf{a} \perp \mathbf{b}$;

(2) Find the value of the real number $k$ if $(\mathbf{a}+2\mathbf{b}) \parallel (2\mathbf{a}-\mathbf{b})$.

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