
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 defined on $\mathbb{R}$: $$f(x)= \frac {1}{2}(\sin \omega x+a\cos \omega x) \quad (a\in \mathbb{R}, 0<\omega\leq1)$$  
It satisfies: $$f(x)=f\left( \frac {\pi}{3}-x\right)$$, and $f(x-\pi)=f(x+\pi)$.  
(Ⅰ) Find the explicit formula for $f(x)$.  
(Ⅱ) Let $x_1$ and $x_2$ be distinct real numbers in the interval $\left(- \frac {\pi}{3}, \frac {5\pi}{3}\right)$, and $f(x_1)=f(x_2)=- \frac {1}{2}$. Find the value of $x_1+x_2$.

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