
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 $f(x)=A\sin (\omega x+\phi)$ $(A > 0,\omega > 0,0\leqslant \phi\leqslant \frac {\pi}{2})$ whose graph passes through point $M(0, \frac {1}{2})$, has the smallest positive period of $\frac {2\pi}{3}$, and has a minimum value of $-1$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Find the intervals of monotonicity for $f(x)$ in the interval $\left[ \frac {\pi}{18}, \frac {5\pi}{9}\right]$.

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