
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 $f(x)=\sin (\omega x+\varphi)$ ($\omega > 0$, $0 < \varphi < \pi$), the distance between two adjacent zeros of the function is $\dfrac{\pi}{2}$. After shifting the graph of $f(x)$ to the left by $\dfrac{\pi}{6}$ units, the corresponding function $g(x)$ is an even function.  
(Ⅰ) Find the analytic expression of the function $f(x)$.  
(Ⅱ) Find the axis of symmetry and the interval of monotonic increase for the function $f(x)$.

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