
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
Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:

- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).

For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 = x_2 - 1 = 3, \quad x_4 = 2x_3 = 6, \quad x_5 = x_4 - 1 = 5, \]
and so on.

The integer \( N = 578 \) can appear in this sequence after the 10th term (e.g., \( x_{12} = 578 \) when \( a = 10 \)), but the integer 579 does not appear in the sequence after the 10th term for any value of \( a \).

What is the smallest integer \( N > 1395 \) that could appear in the sequence after the 10th term for some value of \( a \)?

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