
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, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\frac{2\pi}{3}$ and $b=1$.  
(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle;  
(2) Determine the maximum area of triangle $ABC$.

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