
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 $f\left( x \right)={2}^{x}$, the graph of function $g\left( x \right)$ is symmetric to the graph of function $f\left( x \right)$ about the y-axis.

(1) If $f\left( x \right)=4g\left( x \right)+3$, find the value of $x$.

(2) If there exists $x\in \left[ 0,4 \right]$ such that the inequality $f\left( a+x \right)-g\left( -2x \right)\geqslant 3$ holds, find the range of values for the real number $a$.

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