
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 $\vec{a}=(1,-3)$ and $\vec{b}=(-2,m)$, and $\vec{a} \perp (\vec{a} - \vec{b})$.

   $(1)$ Find the value of the real number $m$;

$(2)$ When $k\vec{a} + \vec{b}$ is parallel to $\vec{a} - \vec{b}$, find the value of the real number $k$.

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