
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 functions $f(x) = x^2 + 4x + a - 5$ and $g(x) = m \cdot 4^{x-1} - 2m + 7$,

(1) If the function $f(x)$ has a zero point in the interval $[-1, 1]$, find the range of the real number $a$;

(2) When $a = 0$, if for any $x_1 \in [1, 2]$, there always exists $x_2 \in [1, 2]$ such that $f(x_1) = g(x_2)$ holds, find the range of the real number $m$;

(3) If the range of $y = f(x), x \in [t, 2]$ is the interval $D$, is there a constant $t$ such that the length of interval $D$ is $6 - 4t$? If it exists, find the value 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}
user