
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
Definition: If a one-variable quadratic equation about $x$, $ax^{2}+bx+c=0 (a\neq 0)$ satisfies $a-b+c=0$, then we call this equation a "golden equation".  
$(1)$ Determine whether the one-variable quadratic equation $2x^{2}+5x+3=0$ is a golden equation, and explain the reason.  
$(2)$ Given that $3x^{2}-ax+b=0$ is a golden equation about $x$, if $a$ is a root of this golden equation, find the value of $a$.

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