
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
Given the following four propositions:
① "k=1" is a necessary and sufficient condition for "the function $y=\cos^2kx-\sin^2kx$ to have a minimum positive period of $\pi$";
② The expression of the function obtained by shifting the graph of $y=\sin(2x- \frac{\pi}{6})$ to the right along the x-axis by $\frac{\pi}{6}$ units is $y=\cos2x$;
③ If the domain of the function $y=\lg(ax^2-2ax+1)$ is $\mathbb{R}$, then the range of the real number $a$ is $(0,1)$;
④ Let O be a point inside $\triangle ABC$, and $\overrightarrow{OA} + \overrightarrow{OC} = -2\overrightarrow{OB}$, then the ratio of the areas of $\triangle AOB$ to $\triangle AOC$ is 1:2;
Among these, the numbers of the true propositions are __________ (write down the numbers of all true propositions).

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