
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 $f(x) = x^4 - 4x^3 + ax^2 - 1$ is monotonically increasing in the interval $[0, 1]$ and monotonically decreasing in the interval $[1, 2]$;  
(1) Find the value of $a$;  
(2) Determine whether there exists a real number $b$ such that the graph of the function $g(x) = bx^2 - 1$ has exactly 2 intersection points with the graph of the function $f(x)$. If it exists, find the value of $b$; if not, explain why.

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