
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 $f(x) = x^2 - 16x + q + 3$  
(1) When $q=1$, find the maximum and minimum values of $f(x)$ on the interval $[-1, 1]$.  
(2) Is there a constant $q$ ($0 < q < 10$) such that the minimum value of $f(x)$ is $-51$ when $x \in [q, 10]$? If it exists, find the value of $q$, otherwise, explain why.

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