
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
Let \( k \) be a fixed positive integer, \( 1 \leqslant k \leqslant n \). Determine the number of integer sequences \( 1 \leqslant a_{1} < a_{2} < \cdots < a_{k} \leqslant n \) such that \( a_{i} \) and \( i \) have the same parity \((i=1, 2, \cdots, k)\).

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