
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
(1) Given $cos(\alpha - \frac{\pi}{3}) = \frac{3}{4}$, find the value of $\sin(\alpha + \frac{7\pi}{6})$.

(2) Given the first four terms of the sequence $\{a_n\}$ are $1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}$, find the general formula for the sequence $\{a_n\}$.

(3) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given $A = 30^{\circ}, b = 2$, if there is only one such triangle, find the range of values for $a$.

(4) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Among the following conditions, which ones can determine that $a = b$? (Fill in the serial numbers)

- $① \sin A = \sin B$
- $② \cos A = \cos B$
- $③ \sin 2A = \sin 2B$
- $④ \cos 2A = \cos 2B$

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