
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 $f(x)=2^{x}$, and the function $h(x)$ is symmetric with respect to the point $(\log_{2}3,\frac{3a}{2})$ with respect to $f(x)$.  
$(1)$ Find the analytical expression of $h(x)$;  
$(2)$ If the equation $f(x)=h(x)$ has two distinct real roots $x_{1}$ and $x_{2}$, and $|x_{1}-x_{2}|=2$, find the value of $a$.

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