
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

Suppose \( a \) and \( b \) are real constants. It is known that for any \( t \in \mathbf{R} \), the graph of the quadratic function in \( x \), given by \( y = \left( t^2 + t + 1 \right) x^2 - 2(a+t)^2 x + t^2 + 3at + b \), always passes through the point \((1,0)\). Determine the values of \( a \) and \( b \).

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