
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 graph of the function $f(x) = ax^4 + bx^2 + c$ passes through the point $(0, 1)$, and the equation of the tangent line at $x = 1$ is $y = x - 2$.
$(1)$ Find the explicit expression for $y = f(x)$;
$(2)$ Find the intervals over which $y = f(x)$ is monotonically increasing.

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