
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 $f(x)=2\sin ωx(0 < ω < 1)$ on $\[0, \dfrac {π}{2}\]$ with a maximum value of $\sqrt {2}$, the graph of $f(x)$ is shifted right by $φ(0 < φ < \dfrac {π}{2})$ units to obtain the function $g(x)$. The graph of $g(x)$ is symmetric about the line $x= \dfrac {7π}{6}$.

(1) Find the analytic expression of function $g(x)$:
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that the first zero point of $g(x)$ on the right side of the $y$-axis is $C$ and $c=4$, find the maximum value of the area $S$ of $\triangle ABC$.

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