
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
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\).

Find the maximum number of matrices verifying these conditions such that any two of them are not similar.

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