
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 the sets $M=\{2,(m^{2}-2m)+(m^{2}+m-2)i\}$ and $P=\{-1,2,4i\}$, find the value of the real number $m$ if $M \cup P = P$.
(2) Given that one root of the equation $x^{2}+4x+a=0\ (a \in \mathbb{R})$ is $x_{1}=-2+i$, find the value of $a$ and the other root of the equation.

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