
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 $f(x) = x|2x-a|$, $g(x) = \frac{x^2-a}{x-1}$ ($a \in \mathbb{R}$).
1. Find the intervals of monotonic increase for the function $f(x)$.
2. If $a < 0$, solve the inequality $f(x) \geq a$.
3. If $0 < a < 12$, and for any $t \in [3, 5]$, the equation $f(x) = g(x)$ always has two distinct real roots when $x \in [3, 5]$, find the range of values for $a$.

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