
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 the diagram below, \(\|\overrightarrow{OA}\| = 2,\) \(\|\overrightarrow{OB}\| = 2,\) and \(\|\overrightarrow{OC}\| = \sqrt{2}.\) Also, \(\tan \angle AOC = 3\) and \(\angle BOC = 45^\circ.\)

[asy]
unitsize(2 cm);

pair A, B, C, O;

A = (2,0);
B = (-1.6,1.6);
C = (0.2,1.4);
O = (0,0);

draw(O--A,Arrow(6));
draw(O--B,Arrow(6));
draw(O--C,Arrow(6));

label("$A$", A, E);
label("$B$", B, NW);
label("$C$", C, N);
label("$O$", O, S);
[/asy]

There exist constants \(m\) and \(n\) so that
\[\overrightarrow{OC} = m \overrightarrow{OA} + n \overrightarrow{OB}.\]Find the ordered pair \((m, n).\)

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