
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
Let the function $f(x) = 2^x$, and the graph of the function $g(x)$ is symmetric to the graph of the function $f(x)$ about the y-axis.  
(1) If $f(x) = 4g(x) + 3$, find the value of $x$;  
(2) If there exists $x \in [0, 4]$ such that the inequality $f(a+x) - g(-2x) \geq 3$ holds, find the range of the real number $a$.

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