
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 angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively. The function $f(A)$ is defined as $f(A) = 2\cos \frac{A}{2}\sin(\pi - \frac{A}{2}) + \sin^2 \frac{A}{2} - \cos^2 \frac{A}{2}$.
1. Determine the maximum value of the function $f(A)$.
2. If $f(A) = 0$, $C = \frac{5\pi}{12}$, and $a = \sqrt{6}$, find the value of $b$.

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