
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 three vectors in the plane $\vec{a}=(3,2)$, $\vec{b}=(-1,2)$, and $\vec{c}=(4,1)$.

(1) Find the real numbers $m$ and $n$ that satisfy $\vec{a}=m\vec{b}+n\vec{c}$;

(2) If $(\vec{a}+k\vec{c}) \parallel (2\vec{b}-\vec{a})$, find the real number $k$.

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