
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 $g(x)=m\cdot4^{x-1}-2m+7$:

(1) If the function $f(x)$ has a zero on 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) Let the range of \(y=f(x)\) where \(x \in [t, 2]\) be 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, 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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