
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 function $f(x) = \sqrt{3}\cos2x + \sin2x$, translate its graph to the right by $\frac{\pi}{6}$ units and then move it up by 1 unit to obtain the graph of function $g(x)$, and determine the minimum value of $a$ for which $|g(x)| \leq a$ always holds.

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