
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 point $\left(m,n\right)$ lies on the parabola $y=ax^{2}+2ax+c$, the following conclusions are made: ① If $a=c$, then the parabola has only one point in common with the $x$-axis; ② If the points $(x_{1}$,$0)$ and $(x_{2}$,$0)$ where the parabola intersects the $x$-axis satisfy $\frac{1}{x_1}+\frac{1}{x_2}=1$, then $c=-2a$; ③ When $-5\leqslant m\leqslant 0$, and the range of $n$ is $-1\leqslant n\leqslant 3$, then the value of $a$ is $\pm \frac{4}{15}$; ④ If the point $\left(m,p\right)$ lies on the line $y=-ax+c-2a$, and when $-2 \lt m \lt -1$, $p \gt n$, then the parabola opens upwards. The correct conclusions are _____. (Fill in the numbers)

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