
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 $f(x) = 2^{x+1}$ defined on $\mathbb{R}$.  
(1) If $f(x)$ can be expressed as the sum of an even function $g(x)$ and an odd function $h(x)$, let $h(x) = t$, $p(t) = g(2x) + 2mh(x) + m^2 - m - 1$ ($m \in \mathbb{R}$), find the expression for $p(t)$.  
(2) If $p(t) \geq m^2 - m - 1$ holds for $x \in [1, 2]$, find the range of values for $m$.

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