
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 a quadratic function $f(x) = ax^2 + bx + c$ ($a, b, c \in \mathbb{R}$), for any real number $x$, the inequality $2x \leq f(x) \leq \frac{1}{2}(x + 1)^2$ always holds.  
(Ⅰ) Find the range of values for $f(-1)$.  
(Ⅱ) Given that for any $x_1, x_2 \in [-3, -1]$, the inequality $|f(x_1) - f(x_2)| \leq 1$ always holds, determine the range of values for $a$.

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