
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 complex number $Z= \frac { \frac {1}{2}}{1+i}+(- \frac {5}{4}+ \frac {9}{4}i)$  
$(1)$ Find the modulus of the complex number $Z$;  
$(2)$ If the complex number $Z$ is a root of the equation $2x^{2}+px+q=0$, find the values of the real numbers $p$ and $q$?

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