
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 $f(x) = \sin x$, the graph of the function $y = f(x)$ is obtained by shortening the abscissa of each point on the graph to $\frac{1}{2}$ of its original length, then shifting it to the left by $\frac{π}{6}$ units, and then shifting it upwards by $\frac{{\sqrt{3}}}{2}$ units, resulting in the graph of $y = g(x)$.  
$(1)$ When $x \in [0, \frac{π}{2}]$, find the range of $g(x)$.  
$(2)$ Given an acute triangle $\triangle ABC$ with the interior angles $A$, $B$, $C$ and the opposite sides $a$, $b$, $c$, if $f(A) = \frac{{\sqrt{3}}}{2}$, $a = 4$, and $b + c = 5$, find the area of $\triangle ABC$.

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