
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 the function f(x) = 2sin(ωx + φ) (-π < φ < 0, ω > 0), its graph is symmetric about the line x = $\frac{\pi}{6}$, and the distance between two adjacent symmetry centers is $\frac{\pi}{2}$.
1. Find the smallest positive period and the monotonically increasing interval of f(x);
2. If the function g(x) = f(x + a) is an even function, find the minimum value of |a|;
3. If the equation f(x) + log$_2$k = 0 has real solutions in the interval [0, $\frac{\pi}{2}$], find the range of real numbers k.

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