
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 a function $f(x)=A\sin (\omega x+\varphi) \ (A > 0,\omega > 0,|\varphi| < \dfrac {\pi}{2})$ with the smallest positive period being $\pi$, the graph of the function is symmetric about the point $( \dfrac {\pi}{12},0)$, and it passes through the point $( \dfrac {\pi}{2},1)$.
$(I)$ Find the analytic expression for the function $f(x)$;
$(II)$ If the equation $2f(x)-a+1=0$ has solutions when $x \in [0, \dfrac {\pi}{2}]$, find the range of the real number $a$.

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