
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 positive integers \( a \) and \( b \), \( a \uparrow \uparrow b \) is defined as follows: \( a \uparrow \uparrow 1 = a \), and \( a \uparrow \uparrow b = a^{a \uparrow \uparrow (b-1)} \) if \( b > 1 \). Find the smallest positive integer \( n \) for which there exists a positive integer \( a \) such that \( a \uparrow \uparrow 6 \not\equiv a \uparrow \uparrow 7 \pmod{n} \).

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