
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 the function $f(x) = ax^{2} - |x-a|$, where $a \in \mathbb{R}$.  
$(1)$ Discuss the parity of the function $f(x)$.  
$(2)$ When $-1 \leq a \leq 1$, if for any $x \in [1,3]$, $f(x) + bx \leq 0$ always holds, find the maximum value of $a^{2} + 3b$.

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