
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $A+C=\frac{2\pi}{3}$, $b=1$.
(1) Let $A=x$, $f(x) = a+c$, find the range of $f(x)$ if $\triangle ABC$ is an acute triangle;
(2) Find the maximum area of $\triangle ABC$.

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