
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 an even function $f(x)$ and an odd function $g(x)$ defined on $R$ satisfying $f(x)+g(x)=2^{x+1}$.  
$(1)$ Find the analytical expressions for $f(x)$ and $g(x)$.  
$(2)$ Given the function $h(x)=af(2x)+g(x)$, if for any $x_{1}$, $x_{2}\in [0,1]$, $|h(x_{1})-h(x_{2})|≤\frac{25}{8}$, find the range of real number $a$.

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