
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
$(1)$ Find the maximum and minimum values of the quadratic function $y=2x^{2}-3x+5$ on the interval $-2\leqslant x\leqslant 2$, and find the corresponding values of $x$.  
$(2)$ Given that the function $y=x^{2}+2ax+1$ has a maximum value of $4$ on the interval $-1\leqslant x\leqslant 2$, find the value of $a$.

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