
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
For the equation about $x$ given by $x^2 + 4x \sin \frac{\alpha}{2} + m \tan \frac{\alpha}{2} = 0$ $(0 < \alpha < \pi)$, there are two identical real roots.

(1) Determine the range of values that the real number $m$ can take.  
(2) If $m + 2 \cos \alpha = \frac{4}{3}$, find the value of $\frac{1 + \sin 2\alpha - \cos 2\alpha}{1 + \tan \alpha}$.

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