
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
If the quadratic function about $x$ is $y=(t+1)x^{2}+(t+2)x+s$ ($t$ and $s$ are constants, $t\neq -1$), and it always has two distinct double value points where the coordinates of the point satisfy $\left(k,2k\right)$, determine the range of $s$.

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