
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
Let the quadratic function $f(x)=ax^2+bx$ $(a\neq 0)$ satisfy the conditions: $(1)$ $f(x)=f(-x-2)$; $(2)$ The graph of the function $f(x)$ is tangent to the line $y=x$.

$(1)$ Find the expression of the function $f(x)$;

$(2)$ If the inequality $\pi^{f(x)} > \left( \frac{1}{\pi}\right)^{2-tx}$ holds for $|t|\leqslant 2$, find the range of the real number $x$.

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