
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 quadratic function \( f(x) = x^2 - 16x + p + 3 \):

(1) If the function has a zero in the interval \([-1, 1]\), find the range of the real number \(p\).

(2) Determine whether there exists a constant \(q (q \geq 0)\) such that when \(x \in [q, 10]\), the range of \(f(x)\) is the interval \(D\), and the length of \(D\) is \(12 - q\). Note: The length of the interval \([a, b] (a < b)\) is \(b - a\).

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