
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
To understand the performance of ninth-grade boys in a school in "throwing a solid ball on their heads with both hands," the school student union randomly selected 50 boys for testing, recorded their throwing distances $x$ (unit: $m$), and grouped and organized them into the following incomplete statistical table. Where group $A$ is $x < 3.5$, group $B$ is $3.5 \leqslant x < 3.9$, group $C$ is $3.9 \leqslant x < 4.7$, group $D$ is $4.7 \leqslant x < 5.5$, group $E$ is $5.5 \leqslant x < 8.5$, group $F$ is $8.5 \leqslant x < 11$, and group $G$ is $x \geqslant 11$.

| Group | Number of Students | Percentage |
|-------|--------------------|------------|
| $A$   |                    | $0.04$     |
| $B$   |                    | $0.10$     |
| $C$   | $8$                |            |
| $D$   | $a$                |            |
| $E$   | $9$                | $0.18$     |
| $F$   |                    | $0.24$     |
| $G$   | $5$                | $b$        |

$(1)$ Fill in the blanks: The value of $a$ is ______; the value of $b$ is ______.

$(2)$ Given the evaluation criteria for the "throwing a solid ball on their heads with both hands" in group $E$ as follows:

| $x$   | $5.5$ | $5.8$ | $6.1$ | $6.4$ | $6.7$ | $7.0$ | $7.3$ | $7.6$ | $7.9$ | $8.2$ | $8.5$ |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| Score | $3.0$ | $3.1$ | $3.2$ | $3.3$ | $3.4$ | $3.5$ | $3.6$ | $3.7$ | $3.8$ | $3.9$ | $4.0$ |

Where the score is recorded as the score of the previous distance if the next distance is not reached. For example, when $x=6.2$, the score is $3.2$.

Among the $9$ students in group $E$, $7$ students scored $3.4$, $3.8$, $3.7$, $3.9$, $3.4$, $3.6$, $3.4$. Estimate the range of the median $m$ of the throwing distances of these 50 boys.

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