
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: $y=ax^{2}+bx+c\left(a \gt 0\right)$, the analytical expression of its graph after a reflection about the $y$-axis is: $y=a\left(x+2\right)^{2}+\left(a-1\right)^{2}$. If the sum of $\left(b+c\right)$ achieves the minimum value, then $a=$____.

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