
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 $f(x)=\sin(x+\frac{π}{6})+\sin(x-\frac{π}{6})+\cos x+a$ has a maximum value of $1$.  
$(1)$ Find the value of $a$;  
$(2)$ Shift the graph of $f(x)$ up by $1$ unit, then shrink the vertical coordinates of all points on the graph to half of the original (keeping the horizontal coordinates unchanged), and shrink the horizontal coordinates of all points on the graph to half of the original (keeping the vertical coordinates unchanged). This results in the graph of a function $g(x)$. In triangle $ABC$, with sides opposite angles $A$, $B$, and $C$ denoted as $a$, $b$, and $c$ respectively, if $g(A)=\frac{1}{2}$ and $a=2$, find the maximum area of triangle $ABC$.

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