
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 $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$. In the interval $[-1,1)$, $f(x)$ is defined as follows:
$$
f(x)=\begin{cases} 
x + a, & -1 \leqslant x < 0, \\
\left| \frac{2}{5} - x \right|, & 0 \leqslant x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left( -\frac{5}{2} \right) = f\left( \frac{9}{2} \right)$, then the value of $f(5a)$ is \_\_\_\_.

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