
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]


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Given $$\overrightarrow {AC}=(\cos \frac {x}{2}+\sin \frac {x}{2}, -\sin \frac {x}{2})$$, $$\overrightarrow {BC}=(\cos \frac {x}{2}-\sin \frac {x}{2}, 2\cos \frac {x}{2})$$, let $f(x) = \overrightarrow {AC}\cdot \overrightarrow {BC}$

(1) Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;  
(2) Suppose the equation $f(x)=a$ has two distinct real roots in the interval $\left[-\frac {\pi}{2}, \frac {\pi}{2}\right]$, find the range of values for $a$.

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