
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) = x|2a - x| + 2x, where a is a real number.
(1) Find the range of values for the real number a such that the function f(x) is an increasing function on R.
(2) If there exists a real number a in [-2, 2] such that the equation f(x) - t f(2a) = 0 has 3 distinct real roots, find the range of values for the real number t.

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