
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
In quadrilateral ABCD on a plane, vector $\overrightarrow{a} = \overrightarrow{AB} = (4,1)$, $\overrightarrow{b} = \overrightarrow{BC} = (3,1)$, $\overrightarrow{c} = \overrightarrow{CD} = (-1,-2)$  
(1) If the vector ($\overrightarrow{a} + 2\overrightarrow{b}$) is perpendicular to the vector ($\overrightarrow{b} - k\overrightarrow{c}$), find the value of the real number $k$;  
(2) If $\overrightarrow{DB} = m\overrightarrow{DA} + n\overrightarrow{DC}$, find the real numbers $m$ and $n$.

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