Problem Description

The deep-blue quantum cryogenic dilution refrigerator is running smoothly. This is the core machine room of the "National Quantum Security Laboratory".

Xiao Lan, the chief architect of the "Quantum 2048" strategic project, is staring at a $2048 \times 2048$ qubit logical array on the screen. This array is the basic unit for building an ultra-large-scale quantum computer. Each bit position must be initialized to one of two states: $L$ (low-energy state) or $Q$ (quantum excited state).

To pass the strict access audit of the "National Quantum Security Laboratory", the logical distribution of the array must strictly follow the following three verification rules derived from quantum entanglement dynamics:

1. Row quantum parity check: In each row of the array, the number of bits in state $Q$ must be odd.
2. Column quantum parity check: In each column of the array, the number of bits in state $Q$ must be odd.
3. Local entanglement constraint check: In any $2 \times 2$ subregion of the array (a grid block of $2$ consecutive adjacent rows and $2$ consecutive adjacent columns), the number of bits in state $Q$ must be odd.

If even one local region fails to meet the rules, the whole chip will undergo quantum decoherence in the superconducting state, causing the computation to fail. Xiao Lan wants to compute how many different initialization filling schemes can pass the audit under the fixed size $2048 \times 2048$.

Now, as Xiao Lan's algorithm consultant, please help him finish the computation. Since the number of schemes may be very large, output the result modulo $998244353$.

Output Format

This is an output-only fill-in-the-blank problem. You only need to compute the result and submit it. The answer is a single integer. When submitting, only fill in this integer; any extra content will not be scored.

Hint

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Translated by ChatGPT 5