
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 $f(x)=|2x+a|+2a$, where $a\in\mathbb{R}$.

(Ⅰ) If for any $x\in\mathbb{R}$, $f(x)$ satisfies $f(x)=f(3-x)$, find the value of $a$;

(Ⅱ) If there exists $x\in\mathbb{R}$ such that $f(x)\leqslant -|2x-1|+a$ holds, find the range of the real number $a$.

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