
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
Given $\sin 2\theta =a$, $\cos 2\theta =b$, $0 \lt \theta\ \ \lt \frac{π}{4}$, provide the five possible answers for the value of $\tan (\theta +\frac{π}{4}$): ① $\frac{b}{{1-a}}$; ② $\frac{a}{{1-b}}$; ③ $\frac{{1+b}}{a}$; ④ $\frac{{1+a}}{b}$; ⑤ $\frac{{a-b+1}}{{a+b-1}}$. The correct answer is ____ (fill in the corresponding number).

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