
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
(1) Given that $α$ and $β$ are the two roots of the equation $2x^{2}+3x+1=0$, find the value of $({{\frac{1}{4}}^{\alpha + \beta }})$;

(2) If ${{x}^{\frac{1}{2}}}+{{x}^{- \frac{1}{2}}}=3$, find the value of $\frac{{{x}^{\frac{3}{2}}}+{{x}^{- \frac{3}{2}}}+2}{x+{{x}^{-1}}+3}$.

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