
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 a function $f(x) = |x-a| + |x-2|$, where $x \in \mathbb{R}$,
(Ⅰ) If the inequality $f(x) \leq a$ has a solution on $\mathbb{R}$, find the minimum value $M$ of the real number $a$.
(Ⅱ) Under the condition of (Ⅰ), given positive real numbers $m$, $n$, and $p$ satisfying $m + 2n + 3p = M$, find the minimum value of $\frac{3}{m} + \frac{2}{n} + \frac{1}{p}$.

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