
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 $f\left(x\right)=4^{x}-2^{x+1}+a$.  
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt a$ with respect to $x$;  
$(2)$ Given the function $g\left(x\right)=f\left(x\right)\cdot [f\left(x+1\right)-2^{2x+1}]+a^{2}$, if the minimum value of $g\left(x\right)$ is $g\left(a\right)$, find the value of $a$ that satisfies $g\left(a\right)=2a$.

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