
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:
$$
f(x) =
\begin{cases}
x + a, & \text{if } -1 \leqslant x < 0, \\
|\frac{2}{5} - x|, & \text{if } 0 \leqslant x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f(-\frac{5}{2}) = f(\frac{9}{2})$, find the value of $f(5a)$.

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