
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 a quadratic function $f(x)$ that satisfies $f(-2-x) = f(-2+x)$ and intersects the $y$-axis at the point $(0,1)$, with a segment of length $2\sqrt{2}$ on the $x$-axis. <br/>$(1)$ Find the analytical expression of the function $f(x)$; <br/>$(2)$ If the function $y = f(x) - ax$ is monotonic on $[2,3]$, find the range of real numbers for $a$; <br/>$(3)$ Solve the inequality $2f(x) - (a+5)x - 2 + a > 0$.

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