
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 the function $f(x)=A(\sin \frac {x}{2}\cos \varphi+\cos \frac {x}{2}\sin \varphi)$ $(A > 0,0 < \varphi < \frac {\pi}{2})$ has a maximum value of $2$, and $f(0)=1$.  
(I) Find the value of $\varphi$;  
(II) Given an acute-angled $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $a=2$, $f(2A)=2$, and $2b\sin C= \sqrt {2}c$, find the area of $\triangle ABC$.

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