
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
Given the function $f(x)=b\cdot a^{x}$ ($a,b$ are constants and $a > 0$, and $a\neq 1$) whose graph passes through points $A(1 , 6)$, $B(3 , 24)$,

$(1)$ Try to determine the analytical expression of the function $f(x)$;

$(2)$ If the inequality about $x$, $\left( \frac{1}{a}\right)^{x}+\left( \frac{1}{b}\right)^{x}-m\geqslant 0$ is always true in the interval $(-\infty,1]$, find the range of values for the real number $m$.

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