
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 the function $f(x)$ be defined on $[-1,0) \cup (0,1]$ as an even function. When $x \in [-1,0)$, $f(x) = x^3 - ax (a \in \mathbb{R})$.

$(1)$ Find the analytical expression for $f(x)$.

$(2)$ Does there exist a real number $a$ such that when $x \in (0,1]$, the maximum value of $f(x)$ is $1$?

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