
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 quadratic equation in \( x \) given by \( a(1+\mathrm{i}) x^{2} + \left(1+a^{2} \mathrm{i}\right) x + a^{2} + \mathrm{i} = 0 \) has real roots, find the value of the real number \( a \).

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