
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 set $A=\{t|t \text{ makes } \{x|x^2+2tx-4t-3 \neq 0\} = \mathbb{R}\}$, and the set $B=\{t|t \text{ makes } \{x|x^2+2tx-2t=0\} \neq \emptyset\}$, where $x$ and $t$ are real numbers.  
(1) Find $A \cap B$;  
(2) Let $m$ be a real number, $g(\alpha)=-\sin^2\alpha+m\cos\alpha-2m$, $\alpha \in [\pi, \frac{3}{2}\pi]$, find $M=\{m|g(\alpha) \in A \cap B\}$.

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