
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 $f(x) = 2^x - \frac{1}{2^{|x|}}$  
$(1)$ If $f(x) = 2$, find the value of $x$;  
$(2)$ If for $t \in [1,2]$, the inequality $2^t f(2t) + mf(t) \geqslant 0$ always holds, find the range of the real number $m$.

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