
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 $a, x \in \mathbb{R}$, the function $$f(x) = \sin 2x - (2 \sqrt {2} + \sqrt {2}a)\sin\left(x+ \frac {\pi}{4}\right) - \frac {2 \sqrt {2}}{\cos\left(x- \frac {\pi}{4}\right)}.$$  
(1) Let $t = \sin x + \cos x$, express the function $f(x)$ as a function $g(t)$ of $t$, and find the expression and domain of $g(t)$;  
(2) For any $x \in \left[0, \frac {\pi}{2}\right]$, if the function $f(x) > -3-2a$ always holds, find the range of values for $a$.

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