
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
Find the explicit formula for $f(x)$ given that:  
- For $x \in (0, 5]$, $x \leq f(x) \leq 2|x-1| + 1$ always holds.  
- When $x \in \mathbb{R}$, the minimum value of $f(x)$ is obtained, and $f(1) = 4a + 1$ holds.  
- Find the value of $a$.  
- Find the largest real number $m$ ($m > 1$) such that there exists a number $t$, only when $x \in [1, m]$, $f(t) \leq \frac{1}{4}(x + t + 1)^2$ holds.

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