
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) = ax^2 + bx + 1 (a > 0), for any x ∈ R, there is always f(x - 4) = f(-x).
1. If the vertex coordinates of the function f(x) are (x0, -3), find the analytical expression of f(x);
2. The minimum value of the function f(x) is recorded as h(a), find the value range of the function H(a) = a • h(a) on a ∈ (0, 4].

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