
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
If the function $f(x)= \begin{cases} a,&x=1 \\ (\frac{1}{2})^{|x-1|}+1,&x\neq 1 \end{cases}$, and the equation $2f^{2}(x)-(2a+3)f(x)+3a=0$ has $5$ distinct real solutions, determine the range of $a$.

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