
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 function $f(x) = m(\sin x + \cos x) - 4\sin x \cos x$, where $x \in [0, \frac{\pi}{2}]$ and $m \in \mathbb{R}$.
1. Let $t = \sin x + \cos x$, with $x \in [0, \frac{\pi}{2}]$. Express $f(x)$ as a function $g(t)$ of $t$, and determine the range of values for $t$.
2. If the inequality $f(x) \geq 0$ holds for all $x \in [0, \frac{\pi}{2}]$, find the range of real numbers $m$.
3. If the equation $f(x) - 2m + 4 = 0$ has real roots on $[0, \frac{\pi}{2}]$, determine the range of real numbers $m$.

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