
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 $\mathbb{R}$, and $f(x)$ can be expressed as the sum of an even function $g(x)$ and an odd function $h(x)$.

$(1)$ Find the expressions for $g(x)$ and $h(x)$;

$(2)$ Let $t=h(x)$, $p(t)=g(2x)+2mh(x)+m^2-m-1$ $(m \in \mathbb{R})$. If $p(t) \geqslant m^2-m-1$ holds for any $x \in [1,2]$, find the range of $m$.

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