
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
(1) Given two non-zero planar vectors $\overrightarrow{a}, \overrightarrow{b}$, with $|\overrightarrow{a}|=2$, and $\overrightarrow{a} \perp (\overrightarrow{a}+2\overrightarrow{b})$, the projection of vector $\overrightarrow{b}$ on the direction of vector $\overrightarrow{a}$ is ______.

(2) For a geometric sequence $\{a_n\}$ with all terms being real numbers, and the sum of the first $n$ terms denoted as $S_n$, given that $S_3=\frac{7}{4}$ and $S_6=\frac{63}{4}$, then $S_9=$ ______.

(3) Given points $F_1, F_2$ are respectively the left and right foci of the ellipse $C: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$, and point $P$ is a point on the ellipse with $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}=0$, $\angle PF_2F_1=5\angle PF_1F_2$, then the eccentricity of the ellipse $C$ is ______.

(4) Given the function $f(x)=(x^2-1)(x^2+ax+b)$ has its graph symmetric about the line $x=3$, then the range of the function $f(x)$ is ______.

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