
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
In the Cartesian coordinate plane, let's call points where the y-coordinate is twice the x-coordinate "ideal points," such as points $(-2, -4)$, $(1, 2)$, $(3, 6)$, etc. There are countless "ideal points."

1. If point $M(a, b)$ is an "ideal point" with a distance of 2 from the y-axis, find the coordinates of point $M$.
2. Does the graph of the function $y = 3mx - 1$ ($m$ is a nonzero constant) contain any "ideal points"? If so, find the coordinates of the "ideal points"; if not, explain why.
3. Given a quadratic function $y = ax^2 + bx + c$ that satisfies the following conditions:
   (i) The graph contains an "ideal point";
   (ii) When $x = 0$ and $x = -2$, the function values are both equal to $5a + 1$.
   If $t = a^2 + a + 1$, find the range of $t$.

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