
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

	 Determine the smallest positive integer  $n \ge 3$  for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where  $A$  denotes the result when the numbers  $2^{10}$ ,  $2^{20}$ ,  $\dots$ ,  $2^{10n}$  are written in decimal notation and concatenated (for example, if  $n=2$  we have  $A = 10241048576$ ).


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