
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 that the function $f(x) = \sin(\omega x + \varphi) + \sqrt{3}\cos(\omega x + \varphi)$ is an odd function, and the distance between two adjacent symmetry axes of the graph of the function $y = f(x)$ is $\frac{\pi}{2}$.

(Ⅰ) Find the value of $f\left(\frac{\pi}{6}\right)$.

(Ⅱ) After shifting the graph of the function $y = f(x)$ to the right by $\frac{\pi}{6}$ units, we obtain the graph of the function $y = g(x)$. Find the interval where the function $g(x)$ is monotonically increasing.

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