
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
For any positive integer \( n \), an \( n \)-tuple of positive integers \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is said to be supersquared if it satisfies both of the following properties:
1. \(x_{1} > x_{2} > x_{3} > \cdots > x_{n}\).
2. The sum \(x_{1}^{2} + x_{2}^{2} + \cdots + x_{k}^{2}\) is a perfect square for each \(1 \leq k \leq n\).

For example, \((12, 9, 8)\) is supersquared, since \(12 > 9 > 8\), and each of \(12^{2}\), \(12^{2} + 9^{2}\), and \(12^{2} + 9^{2} + 8^{2}\) are perfect squares.

(a) Determine all values of \(t\) such that \((32, t, 9)\) is supersquared.

(b) Determine a supersquared 4-tuple \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) with \(x_{1} < 200\).

(c) Determine whether there exists a supersquared 2012-tuple.

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