
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
Define a function $f(x)$ on $\mathbb{R}$ satisfying $f(x+2)=f(x)$, and in the interval $[-1, 1)$, $f(x) = \begin{cases} x-m, & -1 \leq x < 0 \\ |x- \frac{2}{5}|, & 0 \leq x < 1 \end{cases}$, where $m \in \mathbb{R}$. If $f(- \frac{5}{2})=f( \frac{9}{2})$, then $f(5m)=$ \_\_\_\_\_\_.

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