
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 $f(x) = x^2 - 4x + a + 3$, $g(x) = mx + 5 - 2m$.

(I) If $y = f(x)$ has a zero point in the interval $[-1, 1]$, find the range of values for the real number $a$.

(II) When $a = 0$, if for any $x_1 \in [1, 4]$, there always exists $x_2 \in [1, 4]$ such that $f(x_1) = g(x_2)$ holds, find the range of values for the real number $m$.

(III) If the range of the function $y = f(x)$ ($x \in [t, 4]$) is the interval $D$, is there a constant $t$ such that the length of interval $D$ is $7 - 2t$? If it exists, find all values of $t$; if not, please explain why (Note: the length of the interval $[p, q]$ is $q - p$).

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