
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 each integer  $k\ge2$ , the decimal expansions of the numbers  $1024,1024^2,\dots,1024^k$  are concatenated, in that order, to obtain a number  $X_k$ .  (For example,  $X_2 = 10241048576$ .)  If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of  $n$ , where  $n\ge2$  is an integer.

*Proposed by Evan Chen*

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