
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 proposition: If $a$ and $b$ are positive constants, and $a \neq b$, $x$, $y \in (0, +\infty)$, then $\dfrac{a^{2}}{x} + \dfrac{b^{2}}{y} \geqslant \dfrac{(a+b)^{2}}{x+y}$ holds if and only if $\dfrac{a}{x} = \dfrac{b}{y}$, determine the minimum value of the function $f(x) = \dfrac{2}{x} + \dfrac{9}{1-2x} - 5$ and the value of $x$ at which the minimum is achieved ($x \in (0, \dfrac{1}{2})$).

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