
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 proposition $p$: A real number $a$ satisfies the inequality $3^{a} \leqslant 9$, and proposition $q$: The solution set of $x^{2}+3(3-a)x+9 \geqslant 0$ is $\mathbb{R}$. It is known that "$p \land q$" is a true proposition, denoted as condition $r$, and condition $t$: A real number $a$ satisfies $a < m$ or $a > m+ \frac{1}{2}$.  
$(1)$ Find the equivalent condition of condition $r$ (expressed in terms of the range of values of $a$);  
$(2)$ If $r$ is a necessary but not sufficient condition for $\lnot t$, find the value of the positive integer $m$.

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