
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
For the function $f\left(x\right)=\sin |x|+|\cos x|$, the following four statements are made:<br/>①$f\left(x\right)$ is an even function;<br/>②$f\left(x\right)$ is monotonically decreasing on the interval $(\frac{π}{2},π)$;<br/>③The range of $f\left(x\right)$ on the interval $(-\frac{π}{2},\frac{π}{2})$ is $[1,\sqrt{2}]$;<br/>④For $x∈(\frac{5π}{4},\frac{7π}{4})$, $f\left(x\right) \lt 0$ always holds.<br/>The correct statement numbers are ______ (fill in the numbers of all correct statements).

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