
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 $f(x) = 2\sin(2x + \phi) \ (\lvert\phi\rvert < \frac{\pi}{2})$, its graph is shifted to the left by $\frac{\pi}{6}$ units, resulting in an odd function. The function $g(x) = (2 + \sqrt{3})\cos{2x}$ is also given. If the equation $f(x) + g(x) = -2$ has two distinct solutions $\alpha$ and $\beta$ in the interval $[0, \pi)$, find the value of $\cos(\alpha - \beta)$.

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