
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
Let $f(x) = x^2 + 2|x-a| + a$ where $a \in \mathbb{R}$, and $g(x) = \frac{x^2 - 2ax + 1}{x^2}$ where $a \in \mathbb{R}$.  
$(1)$ If the function $f(x)$ is an even function, find the value of the real number $a$ and indicate the monotonic intervals of the function $f(x)$.  
$(2)$ If $a \lt 0$, for all $x_{1} \in [-1, -\frac{1}{3}]$, there exists $x_{2} \in [-2, 2]$ such that $g(x_{1}) = f(x_{2})$. Find the range of values for the real number $a$.

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