
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 power function $$f(x)=(m^{2}+m-1)x^{-2m^{2}+m+3}$$ is increasing on the interval $(0, +\infty)$, $g(x)=-x^2+2|x|+t$, and $h(x)=2^x-2^{-x}$:

(1) Find the value of $m$ and determine the explicit formula for $f(x)$;  
(2) For any $x \in [1, 2]$, there exist $x_1, x_2 \in [1, 2]$ such that $f(x) \leq f(x_1)$, $g(x) \leq g(x_2)$, if $f(x_1) = g(x_2)$, find the value of the real number $t$;  
(3) If $2^xh(2x)+\lambda h(x) \geq 0$ holds for all $x \in [1, 2]$, find the range of the real number $\lambda$.

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