
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 that the parabola $y=ax^{2}+bx+2\left(a\neq 0\right)$ intersects the $x$-axis at points $A(-1,0)$ and $B$, and intersects the $y$-axis at point $C$.<br/>$(1)$ The axis of symmetry of the parabola is the line $x=$______(expressed as an algebraic expression involving $a$);<br/>$(2)$ If $AB=3$, when $x \gt -1$, $y$ increases as $x$ increases, and point $P$ is a point on the parabola below the $x$-axis, and the area of $\triangle BPC$ is divided into $1:2$ by the $x$-axis, then the coordinates of point $P$ are______.

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