
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 $M=\{2, (m^2-2m) + (m^2+m-2)i\}$, $P=\{-1, 2, 4i\}$. If $M \cup P = P$, find the value of the real number $m$.  
(2) Given the equation $x^2+4x+a=0$ ($a \in \mathbb{R}$) has one root $x_1 = -2+i$, 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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