
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 $\triangle ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$ respectively. Which of the following conclusions is (or are) correct:
1. If $a$, $b$, $c$ form an arithmetic sequence, then $B = \frac{\pi}{3}$.
2. If $c = 4$, $b = 2\sqrt{3}$, and $B = \frac{\pi}{6}$, then $\triangle ABC$ has two solutions.
3. If $B = \frac{\pi}{6}$, $b = 1$, and $ac = 2\sqrt{3}$, then $a + c = 2 + \sqrt{3}$.
4. If $(2c - b)\cos A = a \cos B$, then $A = \frac{\pi}{6}$.

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