
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 triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, where $c=2$. Also given are vectors $\overrightarrow{m}=(1,\cos C)$ and $\overrightarrow{n}=(\cos C,1)$ with $\overrightarrow{m}\cdot \overrightarrow{n}=1$.

(1) If $A=45^{\circ}$, find the value of $a$.
(2) If $a+b=4$, find the area of triangle $ABC$.

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