
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 $f(x) = 2x^2 + bx + c$, the solution set of the inequality $f(x) < 0$ is $(0,5)$.

$(1)$ Find the explicit formula of $f(x)$;

$(2)$ If for any $x \in [-1,1]$, the inequality $f(x) + t \leqslant 2$ always holds, find the range of values for $t$.

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