
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
Find all positive integers \( c \) such that there exists a positive constant \( T \) for which
\[
[1, 2, \ldots, n] \leqslant T \cdot 2^{c n}
\]
for any positive integer \( n \). Here, \([1, 2, \ldots, n]\) denotes the least common multiple of \( 1, 2, \ldots, n \).

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