
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 \( x_{0} \), three sequences \(\{x_{n}\}, \{y_{n}\}, \{z_{n}\}\) are defined as follows:
1. \( y_{0} = 4 \) and \( z_{0} = 1 \);
2. if \( x_{n} \) is even for \( n \geq 0 \), then 
\[
x_{n+1} = \frac{x_{n}}{2}, \quad y_{n+1} = 2 y_{n}, \quad z_{n+1} = z_{n};
\]
3. if \( x_{n} \) is odd for \( n \geq 0 \), then 
\[
x_{n+1} = x_{n} - \frac{y_{n}}{2} - z_{n}, \quad y_{n+1} = y_{n}, \quad z_{n+1} = y_{n} + z_{n}.
\]

The integer \( x_{0} \) is said to be good if \( x_{n} = 0 \) for some \( n \geq 1 \). Find the number of good integers less than or equal to 1994.

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