
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
Consider the following four propositions:

1. In a geometric sequence \(\{a_n\}\), the condition \(a_1 > 0\) is necessary for the sequence to be an increasing sequence.
2. Given \(a = \lg 2\), then \(a < a^a < a^{a^a}\).
3. By translating the graph of \(y = 2\tan(x + \frac{\pi}{6})\) to the right by \(\frac{\pi}{6}\) units, and then halving the y-coordinate without altering the x-coordinate, we obtain the graph of \(y = \tan x\).
4. Let \(0 < a < 3\), then the function \(f(x) = x^3 - ax\) for \(0 < x < 1\) has a minimum value but no maximum value.

Identify all correct propositions.

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